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<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">There are three types of arrays in Mathcad: <inlineAttr font-weight="normal">vectors, matrices, and nested matrices</inlineAttr>. A <b>vector</b> is an array of just one row or column of numbers. A <b>matrix</b> is an array with two or more rows and columns. A <b>nested matrix</b> is a matrix or vector with vectors or matrices for each of its entries. This tutorial shows you how to create and work with vectors and matrices, both in their linear algebra context and for iterating over calculations.</p>
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<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">To learn about using arrays to store data for import and export, see <b>
<u>
<link href="./feat_06.xmcd" title="Links to section" popup="false">Handling Data</link>
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<phrase>vectors</phrase>
<phrase>nested matrix</phrase>
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<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">A <b>range variable</b> is quite different from a vector. A range variable is a sequence of real-valued, evenly-spaced numbers. Although range variables can be used as inputs to functions for plotting, typically range variables are used as indices for arrays. Range values must be integers in this case. When a range variable is used in an expression, it produces another range variable as output; you can not access or refer to a particular "entry" in a range, or assign it to a new name. In general, a range variable implies that you want to perform a calculation repeatedly, while a vector is a linear algebra construct or a set of data with an expected set of behaviors for functions such as multiplication, length, and so forth.</p>
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<p style="LIST" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">
<link href="#creating" title="Links to section" popup="false">To Create a Vector or Matrix by Hand</link>
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<link href="#subscripts" title="Links to section" popup="false">Subscripts and Superscripts</link>
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<link href="#range" title="Links to section" popup="false">To Create a Vector or Matrix Using Range Variables</link>
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<link href="#calculating vectors" title="Links to section" popup="false">Linear Algebra with Vectors and Matrices</link>
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<link href="#transpose" title="Links to section" popup="false">Transpose, Inverse, Determinant</link>
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<link href="#functions" title="Links to section" popup="false">Array Functions</link>
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<link href="#iteration" title="Links to section" popup="false">Repeated or Element-wise Calculations with Arrays</link>
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<p style="Heading 2" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">To Create a Vector or Matrix by Hand </p>
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<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">To create a vector or matrix of values defined element by element:</p>
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<b>
<c val="#008080">1.</c>
</b> Type <b>[Ctrl] M</b> or click the <b>Matrix </b>
<inlineAttr font-weight="normal">or </inlineAttr>
<b>Vector</b> button on the <b>Matrix</b> toolbar.</p>
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<b>
<c val="#008080">2.</c>
</b> Enter the dimensions of the vector or matrix you want and click OK or Insert.</p>
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<b>
<c val="#008080">3.</c>
</b> Fill in each empty placeholder with a value. Use the <b>Tab</b> key to move from placeholder to placeholder.</p>
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<p style="directions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">See:</p>
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<rendering item-idref="5"></rendering>
</region>
<region region-id="3391" left="18" top="765" width="45.75" height="10.5" align-x="20.25" align-y="774" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="typing" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">[Ctrl] M</p>
</text>
</region>
<region region-id="2642" left="18" top="894.75" width="131.25" height="14.25" align-x="18" align-y="906" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">After you click OK or Insert:</p>
</text>
</region>
<region region-id="2677" left="18" top="920.25" width="24.75" height="12" align-x="29.25" align-y="930" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="directions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Type:</p>
</text>
</region>
<region region-id="2678" left="150" top="920.25" width="20.25" height="12" align-x="157.5" align-y="930" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="directions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">See:</p>
</text>
</region>
<region region-id="2683" left="150" top="942" width="66" height="73.5" align-x="150" align-y="942" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<picture>
<png item-idref="6" display-width="64.5" display-height="72"></png>
</picture>
<rendering item-idref="7"></rendering>
</region>
<region region-id="2681" left="18" top="945" width="69" height="10.5" align-x="20.25" align-y="954" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="typing" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">1 [Tab] 2 [Tab]</p>
</text>
</region>
<region region-id="3415" left="6" top="1032.75" width="331.5" height="28.5" align-x="15" align-y="1044" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Extend or shorten a matrix by pressing the Insert or Delete buttons while you are already in a matrix.</p>
</text>
</region>
<region region-id="3416" left="0" top="1086" width="6000" height="6" align-x="0" align-y="1086" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<pageBreak></pageBreak>
</region>
<region region-id="3393" left="6" top="1094.25" width="56.25" height="12" align-x="27" align-y="1104" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="false">
<p style="Heading 2" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Examples </p>
</text>
</region>
<region region-id="2692" left="6" top="1110.75" width="261" height="14.25" align-x="16.5" align-y="1122" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Here are several arrays with various types of data in them:</p>
</text>
</region>
<region region-id="3529" left="18" top="1140.75" width="82.5" height="14.25" align-x="20.25" align-y="1152" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Comment" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">1 row, 3 columns</p>
</text>
</region>
<region region-id="3521" left="156" top="1140.75" width="71.25" height="15.75" align-x="190.5" align-y="1152" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:matrix rows="1" cols="3">
<ml:real>3</ml:real>
<ml:real>5.2</ml:real>
<ml:real>10.1</ml:real>
</ml:matrix>
</math>
<rendering item-idref="8"></rendering>
</region>
<region region-id="3528" left="18" top="1188.75" width="82.5" height="14.25" align-x="20.25" align-y="1200" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Comment" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">3 rows, 1 column</p>
</text>
</region>
<region region-id="3520" left="156" top="1167.75" width="45" height="57.75" align-x="177.75" align-y="1200" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:matrix rows="3" cols="1">
<ml:imag symbol="i">5</ml:imag>
<ml:apply>
<ml:plus></ml:plus>
<ml:real>1</ml:real>
<ml:imag symbol="i">2</ml:imag>
</ml:apply>
<ml:real>6</ml:real>
</ml:matrix>
</math>
<rendering item-idref="9"></rendering>
</region>
<region region-id="3527" left="18" top="1278.75" width="83.25" height="14.25" align-x="20.25" align-y="1290" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Comment" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">3 rows, 3 columns</p>
</text>
</region>
<region region-id="3519" left="156" top="1269.75" width="51.75" height="57.75" align-x="180.75" align-y="1302" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:matrix rows="3" cols="3">
<ml:real>1</ml:real>
<ml:real>4</ml:real>
<ml:real>7</ml:real>
<ml:real>2</ml:real>
<ml:real>5</ml:real>
<ml:real>8</ml:real>
<ml:real>3</ml:real>
<ml:real>6</ml:real>
<ml:real>9</ml:real>
</ml:matrix>
</math>
<rendering item-idref="10"></rendering>
</region>
<region region-id="3526" left="18" top="1376.25" width="117" height="36" align-x="29.25" align-y="1386" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="directions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Click on 9 in the matrix above and select <b>Insert &gt; Matrix</b> then <b>Insert</b>.</p>
</text>
</region>
<region region-id="3518" left="156" top="1376.25" width="20.25" height="12" align-x="156" align-y="1386" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="directions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">See:</p>
</text>
</region>
<region region-id="3517" left="156" top="1394.25" width="92.25" height="120.75" align-x="201" align-y="1458" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:matrix rows="6" cols="6">
<ml:real>1</ml:real>
<ml:real>4</ml:real>
<ml:real>7</ml:real>
<ml:placeholder></ml:placeholder>
<ml:placeholder></ml:placeholder>
<ml:placeholder></ml:placeholder>
<ml:real>2</ml:real>
<ml:real>5</ml:real>
<ml:real>8</ml:real>
<ml:placeholder></ml:placeholder>
<ml:placeholder></ml:placeholder>
<ml:placeholder></ml:placeholder>
<ml:real>3</ml:real>
<ml:real>6</ml:real>
<ml:real>9</ml:real>
<ml:placeholder></ml:placeholder>
<ml:placeholder></ml:placeholder>
<ml:placeholder></ml:placeholder>
<ml:placeholder></ml:placeholder>
<ml:placeholder></ml:placeholder>
<ml:placeholder></ml:placeholder>
<ml:placeholder></ml:placeholder>
<ml:placeholder></ml:placeholder>
<ml:placeholder></ml:placeholder>
<ml:placeholder></ml:placeholder>
<ml:placeholder></ml:placeholder>
<ml:placeholder></ml:placeholder>
<ml:placeholder></ml:placeholder>
<ml:placeholder></ml:placeholder>
<ml:placeholder></ml:placeholder>
<ml:placeholder></ml:placeholder>
<ml:placeholder></ml:placeholder>
<ml:placeholder></ml:placeholder>
<ml:placeholder></ml:placeholder>
<ml:placeholder></ml:placeholder>
<ml:placeholder></ml:placeholder>
</ml:matrix>
</math>
<rendering item-idref="11"></rendering>
</region>
<region region-id="3539" left="18" top="1532.25" width="113.25" height="36" align-x="29.25" align-y="1542" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="directions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Click on a placeholder and select <b>Insert &gt; Matrix</b> then <b>Delete</b>.</p>
</text>
</region>
<region region-id="3540" left="156" top="1532.25" width="20.25" height="12" align-x="156" align-y="1542" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="false">
<p style="directions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">See:</p>
</text>
</region>
<region region-id="3541" left="234" top="1530.75" width="93.75" height="57" align-x="234" align-y="1542" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">
<b>
<c val="#008080">Note</c>
</b>: Depending where you click you might see a different result.</p>
</text>
</region>
<region region-id="3515" left="156" top="1545.75" width="51.75" height="57.75" align-x="180.75" align-y="1578" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:matrix rows="3" cols="3">
<ml:real>1</ml:real>
<ml:real>4</ml:real>
<ml:real>7</ml:real>
<ml:real>2</ml:real>
<ml:real>5</ml:real>
<ml:real>8</ml:real>
<ml:real>3</ml:real>
<ml:real>6</ml:real>
<ml:real>9</ml:real>
</ml:matrix>
</math>
<rendering item-idref="12"></rendering>
</region>
<region region-id="3482" left="24" top="1638.75" width="87.75" height="14.25" align-x="26.25" align-y="1650" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Comment" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">2 rows, 3 columns</p>
</text>
</region>
<region region-id="3514" left="156" top="1628.25" width="120.75" height="36.75" align-x="215.25" align-y="1650" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:matrix rows="2" cols="3">
<ml:str xml:space="preserve">dog</ml:str>
<ml:real>1</ml:real>
<ml:str xml:space="preserve">cat</ml:str>
<ml:real>3</ml:real>
<ml:str xml:space="preserve">mouse</ml:str>
<ml:real>2</ml:real>
</ml:matrix>
</math>
<rendering item-idref="13"></rendering>
</region>
<region region-id="3437" left="0" top="1674" width="6000" height="6" align-x="0" align-y="1674" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<pageBreak></pageBreak>
</region>
<region region-id="2788" left="6" top="1680.75" width="341.25" height="85.5" align-x="17.25" align-y="1692" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">This last example contains strings, or words, and numeric values. It is helpful to be able to mix data in an array, particularly if you are copying and pasting that data from another application. In fact, data copied and pasted into matrices (from tabular applications such as Excel or an ASCII worksheet that contains rows and columns) preserves its characteristics, including text, numerics, complex numbers, or empty cells. </p>
</text>
</region>
<region region-id="2789" left="6" top="1788.75" width="341.25" height="57" align-x="13.5" align-y="1800" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Try to create these arrays yourself. To enter a string, type [<b>"</b>] in the placeholder and then enter the text. From within the string quotes, you cannot press <b>[Tab]</b> to move to the next placeholder. You can use the arrow keys to enter the next entry or click on the placeholder.</p>
</text>
</region>
<region region-id="3456" left="6" top="1854.75" width="27.75" height="14.25" align-x="13.5" align-y="1866" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Try it!</p>
</text>
</region>
<region region-id="2766" left="6" top="2028.75" width="341.25" height="28.5" align-x="15" align-y="2040" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">You can also assign a variable to a vector or matrix in order to use it later in the worksheet for calculations.</p>
</text>
</region>
<region region-id="3450" left="0" top="2058" width="6000" height="6" align-x="0" align-y="2058" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<pageBreak></pageBreak>
</region>
<region region-id="3461" left="6" top="2072.25" width="178.5" height="12" align-x="24" align-y="2082" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="false">
<p style="Heading 2" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">To Copy Excel Cells into Mathcad</p>
</text>
</region>
<region region-id="3448" left="6" top="2088.75" width="433.5" height="14.25" align-x="6" align-y="2100" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">This image shows Excel cells that you might copy to Mathcad.</p>
</text>
</region>
<region region-id="3445" left="24" top="2118" width="194.25" height="66" align-x="24" align-y="2118" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<picture>
<png item-idref="14" display-width="192.75" display-height="64.5"></png>
</picture>
<rendering item-idref="15"></rendering>
</region>
<region region-id="3449" left="6" top="2196.75" width="341.25" height="28.5" align-x="6" align-y="2208" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">If you place the red crosshair in a blank region of the worksheet and choose <b>Edit &gt; Paste Special</b> as "Unformatted Text As Number," you get:</p>
</text>
</region>
<region region-id="3447" left="24" top="2237.25" width="208.5" height="115.5" align-x="127.5" align-y="2298" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:matrix rows="5" cols="4">
<ml:str xml:space="preserve">temp</ml:str>
<ml:real>98</ml:real>
<ml:real>98.2</ml:real>
<ml:real>98.6</ml:real>
<ml:real>98.7</ml:real>
<ml:str xml:space="preserve">time</ml:str>
<ml:real>1</ml:real>
<ml:real>3</ml:real>
<ml:real>4</ml:real>
<ml:real>6</ml:real>
<ml:str xml:space="preserve">phase</ml:str>
<ml:imag symbol="i">0.2</ml:imag>
<ml:apply>
<ml:plus></ml:plus>
<ml:real>0.2</ml:real>
<ml:imag symbol="i">0.5</ml:imag>
</ml:apply>
<ml:apply>
<ml:plus></ml:plus>
<ml:real>0.3</ml:real>
<ml:imag symbol="i">0.04</ml:imag>
</ml:apply>
<ml:real>0.6</ml:real>
<ml:str xml:space="preserve">Amplitude</ml:str>
<ml:real>700</ml:real>
<ml:apply>
<ml:mult style="auto-select"></ml:mult>
<ml:real>9</ml:real>
<ml:apply>
<ml:pow></ml:pow>
<ml:real>10</ml:real>
<ml:real>3</ml:real>
</ml:apply>
</ml:apply>
<ml:apply>
<ml:mult style="auto-select"></ml:mult>
<ml:real>1.1</ml:real>
<ml:apply>
<ml:pow></ml:pow>
<ml:real>10</ml:real>
<ml:real>4</ml:real>
</ml:apply>
</ml:apply>
<ml:apply>
<ml:mult style="auto-select"></ml:mult>
<ml:real>1.6</ml:real>
<ml:apply>
<ml:pow></ml:pow>
<ml:real>10</ml:real>
<ml:real>4</ml:real>
</ml:apply>
</ml:apply>
</ml:matrix>
</math>
<rendering item-idref="16"></rendering>
</region>
<region region-id="3451" left="6" top="2376.75" width="27.75" height="14.25" align-x="13.5" align-y="2388" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Try it!</p>
</text>
</region>
<region region-id="3453" left="6" top="2396.25" width="264" height="12" align-x="17.25" align-y="2406" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="false">
<p style="directions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Type some numbers into Excel, then copy them into Mathcad.</p>
</text>
</region>
<region region-id="3457" left="0" top="2580" width="6000" height="6" align-x="0" align-y="2580" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<pageBreak></pageBreak>
</region>
<region region-id="3460" left="6" top="2588.25" width="130.5" height="12" align-x="30" align-y="2598" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="false">
<p style="Heading 2" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">To Compute with Arrays</p>
</text>
</region>
<region region-id="2781" left="6" top="2604.75" width="341.25" height="28.5" align-x="9.75" align-y="2616" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">If you define <inlineAttr font-style="normal">x </inlineAttr>to be a 1 <f family="Arial" charset="0" size="10">x</f> 3 vector and <inlineAttr font-style="normal">y</inlineAttr> to be a 3 <f family="Arial" charset="0" size="10">x</f> 1 vector, then you can use those definitions to compute the dot product of the two vectors:</p>
</text>
</region>
<region region-id="2782" left="18" top="2664.75" width="76.5" height="15.75" align-x="31.5" align-y="2676" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:define>
<ml:id xml:space="preserve">x</ml:id>
<ml:matrix rows="1" cols="3">
<ml:real>2</ml:real>
<ml:real>7</ml:real>
<ml:real>10</ml:real>
</ml:matrix>
</ml:define>
</math>
<rendering item-idref="17"></rendering>
</region>
<region region-id="2783" left="126" top="2643.75" width="46.5" height="57.75" align-x="139.5" align-y="2676" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:define>
<ml:id xml:space="preserve">y</ml:id>
<ml:matrix rows="3" cols="1">
<ml:real>4</ml:real>
<ml:real>1</ml:real>
<ml:real>5</ml:real>
</ml:matrix>
</ml:define>
</math>
<rendering item-idref="18"></rendering>
</region>
<region region-id="2779" left="18" top="2706.75" width="53.25" height="15.75" align-x="43.5" align-y="2718" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:eval placeholderMultiplicationStyle="default" resultMultiplicationStyle="default">
<ml:apply>
<ml:mult style="dot"></ml:mult>
<ml:id xml:space="preserve">x</ml:id>
<ml:id xml:space="preserve">y</ml:id>
</ml:apply>
<result xmlns="http://schemas.mathsoft.com/math30">
<ml:real>65</ml:real>
</result>
</ml:eval>
</math>
<rendering item-idref="19"></rendering>
</region>
<region region-id="3553" left="6" top="2730.75" width="161.25" height="14.25" align-x="18.75" align-y="2742" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="false">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Many functions take arrays, as well:</p>
</text>
</region>
<region region-id="3554" left="18" top="2748.75" width="70.5" height="15.75" align-x="66.75" align-y="2760" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:eval placeholderMultiplicationStyle="default" resultMultiplicationStyle="default">
<ml:apply>
<ml:id xml:space="preserve">length</ml:id>
<ml:id xml:space="preserve">y</ml:id>
</ml:apply>
<result xmlns="http://schemas.mathsoft.com/math30">
<ml:real>3</ml:real>
</result>
</ml:eval>
</math>
<rendering item-idref="20"></rendering>
</region>
<region region-id="3556" left="126" top="2748.75" width="60" height="15.75" align-x="164.25" align-y="2760" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:eval placeholderMultiplicationStyle="default" resultMultiplicationStyle="default">
<ml:apply>
<ml:id xml:space="preserve">min</ml:id>
<ml:id xml:space="preserve">y</ml:id>
</ml:apply>
<result xmlns="http://schemas.mathsoft.com/math30">
<ml:real>1</ml:real>
</result>
</ml:eval>
</math>
<rendering item-idref="21"></rendering>
</region>
<region region-id="3557" left="228" top="2727.75" width="100.5" height="57.75" align-x="262.5" align-y="2760" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:eval placeholderMultiplicationStyle="default" resultMultiplicationStyle="default">
<ml:apply>
<ml:id xml:space="preserve">sin</ml:id>
<ml:id xml:space="preserve">y</ml:id>
</ml:apply>
<result xmlns="http://schemas.mathsoft.com/math30">
<ml:matrix rows="3" cols="1">
<ml:real>-0.7568024953079282</ml:real>
<ml:real>0.8414709848078965</ml:real>
<ml:real>-0.95892427466313845</ml:real>
</ml:matrix>
</result>
</ml:eval>
</math>
<rendering item-idref="22"></rendering>
</region>
<region region-id="2866" left="6" top="2792.25" width="81.75" height="12" align-x="6" align-y="2802" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="subscripts">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Heading 2" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Subscripts</p>
</text>
<indexes>
<phrase>subscripts, vector and matrix</phrase>
<phrase>array subscripts</phrase>
</indexes>
</region>
<region region-id="2867" left="6" top="2808.75" width="319.5" height="57" align-x="6" align-y="2820" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">
<b>Subscripts</b> are the numbers or variables that identify a single element or a sequence of elements in an array. Subscripts are written below an array variable. To type an array subscript, type <f family="Tahoma" charset="0" size="9">
<b>[</b>
</f> or click the <b>Subscript</b> button, shown at the right, on the <b>Matrix</b> toolbar. </p>
</text>
</region>
<region region-id="3057" left="330" top="2850" width="16.5" height="16.5" align-x="330" align-y="2850" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<picture>
<png item-idref="23" display-width="15" display-height="15"></png>
</picture>
<rendering item-idref="24"></rendering>
</region>
<region region-id="3056" left="6" top="2880.75" width="285" height="14.25" align-x="6" align-y="2892" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="false">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">A subscript on a <b>vector</b> refers to the <b>nth element</b> of a vector. </p>
</text>
</region>
<region region-id="2868" left="18" top="2904.75" width="14.25" height="17.25" align-x="24" align-y="2916" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:apply>
<ml:indexer></ml:indexer>
<ml:id xml:space="preserve">V</ml:id>
<ml:id xml:space="preserve">n</ml:id>
</ml:apply>
</math>
<rendering item-idref="25"></rendering>
</region>
<region region-id="2869" left="84" top="2904.75" width="77.25" height="14.25" align-x="95.25" align-y="2916" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Comment" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Read, "V sub n."</p>
</text>
</region>
<region region-id="2870" left="6" top="2928.75" width="341.25" height="28.5" align-x="6" align-y="2940" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">A subscript on a <b>matrix</b> refers to an element in the <b>mth row</b> and <b>nth column</b> in the matrix. </p>
</text>
</region>
<region region-id="2871" left="18" top="2964.75" width="28.5" height="17.25" align-x="31.5" align-y="2976" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:apply>
<ml:indexer></ml:indexer>
<ml:id xml:space="preserve">A</ml:id>
<ml:sequence>
<ml:id xml:space="preserve">m</ml:id>
<ml:id xml:space="preserve">n</ml:id>
</ml:sequence>
</ml:apply>
</math>
<rendering item-idref="26"></rendering>
</region>
<region region-id="2874" left="6" top="2988.75" width="316.5" height="45" align-x="6" align-y="3000" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">By default in Mathcad, the first element has the index 0. This means that the first row or column of an array is always denoted with the subscript <region region-id="368" left="49.5" top="3018" width="6.75" height="15.75" align-x="51.75" align-y="3029.25" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:real>0</ml:real>
</math>
<rendering item-idref="27"></rendering>
</region>. </p>
</text>
</region>
<region region-id="3046" left="0" top="3048" width="6000" height="6" align-x="0" align-y="3048" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<pageBreak></pageBreak>
</region>
<region region-id="3399" left="6" top="3056.25" width="56.25" height="12" align-x="6" align-y="3066" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="false">
<p style="Heading 2" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Examples </p>
</text>
</region>
<region region-id="2876" left="6" top="3072.75" width="163.5" height="14.25" align-x="16.5" align-y="3084" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">Here is a vector with three elements:</p>
</text>
</region>
<region region-id="2877" left="18" top="3099.75" width="63" height="57.75" align-x="42" align-y="3132" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:define>
<ml:id xml:space="preserve">vec</ml:id>
<ml:matrix rows="3" cols="1">
<ml:real>7</ml:real>
<ml:real>56</ml:real>
<ml:real>5</ml:real>
</ml:matrix>
</ml:define>
</math>
<rendering item-idref="28"></rendering>
</region>
<region region-id="2878" left="6" top="3168.75" width="368.25" height="14.25" align-x="6" align-y="3180" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">If you type <f family="Tahoma" charset="0" size="9">
<b>vec[0=</b>
</f> you get the first element of the vector:</p>
</text>
</region>
<region region-id="2879" left="18" top="3198.75" width="50.25" height="17.25" align-x="46.5" align-y="3210" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:eval placeholderMultiplicationStyle="default" resultMultiplicationStyle="default">
<ml:apply>
<ml:indexer></ml:indexer>
<ml:id xml:space="preserve">vec</ml:id>
<ml:real>0</ml:real>
</ml:apply>
<result xmlns="http://schemas.mathsoft.com/math30">
<ml:real>7</ml:real>
</result>
</ml:eval>
</math>
<rendering item-idref="29"></rendering>
</region>
<region region-id="2880" left="6" top="3228.75" width="318" height="14.25" align-x="6" align-y="3240" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Likewise, if you type <f family="Tahoma" charset="0" size="9">
<b>v</b>
</f>
<f family="Tahoma" charset="0" size="9">
<b>ec[1=</b>
</f> you get the second element of the vector:</p>
</text>
</region>
<region region-id="2881" left="18" top="3264.75" width="56.25" height="17.25" align-x="46.5" align-y="3276" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:eval placeholderMultiplicationStyle="default" resultMultiplicationStyle="default">
<ml:apply>
<ml:indexer></ml:indexer>
<ml:id xml:space="preserve">vec</ml:id>
<ml:real>1</ml:real>
</ml:apply>
<result xmlns="http://schemas.mathsoft.com/math30">
<ml:real>56</ml:real>
</result>
</ml:eval>
</math>
<rendering item-idref="30"></rendering>
</region>
<region region-id="2883" left="6" top="3300.75" width="99.75" height="14.25" align-x="16.5" align-y="3312" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Here is a 3 <f family="Arial" charset="0" size="10">x</f> 3 matrix:</p>
</text>
</region>
<region region-id="2884" left="18" top="3327.75" width="86.25" height="57.75" align-x="36.75" align-y="3360" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:define>
<ml:id xml:space="preserve">M</ml:id>
<ml:matrix rows="3" cols="3">
<ml:real>7</ml:real>
<ml:real>5</ml:real>
<ml:real>56</ml:real>
<ml:real>6</ml:real>
<ml:real>5</ml:real>
<ml:real>8</ml:real>
<ml:real>5</ml:real>
<ml:real>5</ml:real>
<ml:real>9</ml:real>
</ml:matrix>
</ml:define>
</math>
<rendering item-idref="31"></rendering>
</region>
<region region-id="2885" left="6" top="3402.75" width="341.25" height="14.25" align-x="6" align-y="3414" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">If you type <f family="Tahoma" charset="0" size="9">
<b>M[0,1=</b>
</f> you get the element in the first row and second column:</p>
</text>
</region>
<region region-id="2889" left="18" top="3444.75" width="56.25" height="17.25" align-x="52.5" align-y="3456" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:eval placeholderMultiplicationStyle="default" resultMultiplicationStyle="default">
<ml:apply>
<ml:indexer></ml:indexer>
<ml:id xml:space="preserve">M</ml:id>
<ml:sequence>
<ml:real>0</ml:real>
<ml:real>1</ml:real>
</ml:sequence>
</ml:apply>
<result xmlns="http://schemas.mathsoft.com/math30">
<ml:real>6</ml:real>
</result>
</ml:eval>
</math>
<rendering item-idref="32"></rendering>
</region>
<region region-id="3059" left="6" top="3468.75" width="323.25" height="71.25" align-x="6" align-y="3480" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">You can change the default indexing of arrays in any worksheet by changing the ORIGIN under <b>Tools &gt; Worksheet Options</b>
<inlineAttr font-weight="normal"> or by redefining the built-in variable ORIGIN to 1 in your worksheet. It is useful to make a global change if you wish to start your arrays at 1. You can set this value in a template if you want it for all new worksheets.</inlineAttr>
</p>
</text>
<indexes>
<phrase>ORIGIN of an array</phrase>
<phrase>array origin</phrase>
<phrase>matrix indexing</phrase>
</indexes>
</region>
<region region-id="3060" left="6" top="3554.25" width="137.25" height="12" align-x="6" align-y="3564" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Heading 2" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Superscripts</p>
</text>
<indexes>
<phrase>superscripts, array</phrase>
<phrase>array superscripts</phrase>
<phrase>columns in arrays</phrase>
<phrase>column operator</phrase>
</indexes>
</region>
<region region-id="2898" left="6" top="3570.75" width="341.25" height="57" align-x="6" align-y="3582" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">The superscript of an array refers to a column of the array. For example, if you want to reference an entire column of a particular array, type <b>[Ctrl] 6</b> to insert a superscript or click the <b>Matrix Column</b> button on the <b>Matrix</b> toolbar. </p>
</text>
</region>
<region region-id="3047" left="0" top="3636" width="6000" height="6" align-x="0" align-y="3636" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<pageBreak></pageBreak>
</region>
<region region-id="3401" left="6" top="3650.25" width="50.25" height="12" align-x="6" align-y="3660" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="false">
<p style="Heading 2" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Example </p>
</text>
</region>
<region region-id="3065" left="6" top="3672.75" width="341.25" height="14.25" align-x="6" align-y="3684" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">If you type <f family="Tahoma" charset="0" size="9">
<b>M[Ctrl]60=</b>
</f>, you get the first column of the matrix <b>M</b>:</p>
</text>
</region>
<region region-id="3063" left="24" top="3699.75" width="90.75" height="57.75" align-x="42" align-y="3732" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:eval placeholderMultiplicationStyle="default" resultMultiplicationStyle="default">
<ml:id xml:space="preserve">M</ml:id>
<result xmlns="http://schemas.mathsoft.com/math30">
<ml:matrix rows="3" cols="3">
<ml:real>7</ml:real>
<ml:real>5</ml:real>
<ml:real>56</ml:real>
<ml:real>6</ml:real>
<ml:real>5</ml:real>
<ml:real>8</ml:real>
<ml:real>5</ml:real>
<ml:real>5</ml:real>
<ml:real>9</ml:real>
</ml:matrix>
</result>
</ml:eval>
</math>
<rendering item-idref="33"></rendering>
</region>
<region region-id="3064" left="132" top="3699.75" width="75.75" height="57.75" align-x="163.5" align-y="3732" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:eval placeholderMultiplicationStyle="default" resultMultiplicationStyle="default">
<ml:apply>
<ml:matcol></ml:matcol>
<ml:id xml:space="preserve">M</ml:id>
<ml:real>0</ml:real>
</ml:apply>
<result xmlns="http://schemas.mathsoft.com/math30">
<ml:matrix rows="3" cols="1">
<ml:real>7</ml:real>
<ml:real>5</ml:real>
<ml:real>56</ml:real>
</ml:matrix>
</result>
</ml:eval>
</math>
<rendering item-idref="34"></rendering>
</region>
<region region-id="2505" left="6" top="3782.25" width="324" height="12" align-x="6" align-y="3792" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="range">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Heading 2" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">To Create a Vector or Matrix Using Range Variables</p>
</text>
<indexes>
<phrase>range variables, vector and matrix</phrase>
</indexes>
</region>
<region region-id="2796" left="6" top="3798.75" width="341.25" height="42.75" align-x="6" align-y="3810" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Another way to define vectors and matrices is to create a definition dependent on a range variable or variables, which serve as the indices for the array.</p>
</text>
</region>
<region region-id="2842" left="6" top="3846.75" width="341.25" height="128.25" align-x="6" align-y="3858" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="List_custom" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">
<b>
<c val="#008080">1.</c>
</b> Define a range variable from 1 to <b>n</b>
<inlineAttr font-weight="normal">,</inlineAttr> where <inlineAttr font-weight="normal">n</inlineAttr> is the number of rows you want in the vector. For a matrix, define a second range variable from 1 to <b>m</b>
<inlineAttr font-weight="normal">,</inlineAttr> where <inlineAttr font-weight="normal">m</inlineAttr> is the number of columns.</p>
<p style="List_custom" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">
<b>
<c val="#008080">2.</c>
</b> Now, define your array, <b>A</b>. On the left side of your array definition, type <f family="Tahoma" charset="0" size="9">
<b>A[n</b>
</f> for a vector and <f family="Tahoma" charset="0" size="9">
<b>A[n,m</b>
</f> for a matrix. Remember that the open bracket is the keystroke for the array subscript.</p>
<p style="List_custom" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">
<c val="#008080">
<b>3.</b>
</c> On the right side of your array definition, type an expression in terms of <inlineAttr font-weight="normal">n</inlineAttr> for a vector and <inlineAttr font-weight="normal">n</inlineAttr> and <inlineAttr font-weight="normal">m</inlineAttr> for a matrix.</p>
<p style="List_custom" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">
<b>
<c val="#008080">4.</c>
</b> To view the array you have defined, type <f family="Tahoma" charset="0" size="9">
<b>A= </b>
</f>below the definition.</p>
</text>
</region>
<region region-id="3025" left="0" top="3990" width="6000" height="6" align-x="0" align-y="3990" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<pageBreak></pageBreak>
</region>
<region region-id="3403" left="6" top="4010.25" width="50.25" height="12" align-x="6" align-y="4020" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="false">
<p style="Heading 2" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Example </p>
</text>
</region>
<region region-id="2508" left="6" top="4026.75" width="240" height="14.25" align-x="6" align-y="4038" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">First, define a range variable, <b>i</b>, by typing <f family="Tahoma" charset="0" size="9">
<b>i:0;2</b>
</f>. </p>
</text>
</region>
<region region-id="2801" left="18" top="4050.75" width="42.75" height="15.75" align-x="28.5" align-y="4062" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:define>
<ml:id xml:space="preserve">i</ml:id>
<ml:range>
<ml:real>0</ml:real>
<ml:real>2</ml:real>
</ml:range>
</ml:define>
</math>
<rendering item-idref="35"></rendering>
</region>
<region region-id="2510" left="6" top="4080.75" width="341.25" height="28.5" align-x="6" align-y="4092" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Now, type <f family="Tahoma" charset="0" size="9">
<b>B[i:i+4</b>
</f>. You have assigned to each element in the vector B the row number plus 4, that is the values 4, 5, and 6.</p>
</text>
</region>
<region region-id="2800" left="18" top="4140.75" width="50.25" height="17.25" align-x="36" align-y="4152" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:define>
<ml:apply>
<ml:indexer></ml:indexer>
<ml:id xml:space="preserve">B</ml:id>
<ml:id xml:space="preserve">i</ml:id>
</ml:apply>
<ml:apply>
<ml:plus></ml:plus>
<ml:id xml:space="preserve">i</ml:id>
<ml:real>4</ml:real>
</ml:apply>
</ml:define>
</math>
<rendering item-idref="36"></rendering>
</region>
<region region-id="2802" left="102" top="4119.75" width="53.25" height="57.75" align-x="117" align-y="4152" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:eval placeholderMultiplicationStyle="default" resultMultiplicationStyle="default">
<ml:id xml:space="preserve">B</ml:id>
<result xmlns="http://schemas.mathsoft.com/math30">
<ml:matrix rows="3" cols="1">
<ml:real>4</ml:real>
<ml:real>5</ml:real>
<ml:real>6</ml:real>
</ml:matrix>
</result>
</ml:eval>
</math>
<rendering item-idref="37"></rendering>
</region>
<region region-id="2803" left="6" top="4188.75" width="341.25" height="42.75" align-x="6" align-y="4200" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">
<b>
<c val="#008080">Note</c>
</b>: In Mathcad, all vectors defined by a range variable are column vectors, that is, vectors that go down, not across. Column vectors are expected by most of the vector functions in Mathcad.</p>
</text>
</region>
<region region-id="3559" left="6" top="4244.25" width="172.5" height="12" align-x="6" align-y="4254" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="false">
<p style="Heading 2" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Range Variables versus Vectors</p>
</text>
</region>
<region region-id="3561" left="6" top="4260.75" width="270.75" height="14.25" align-x="6" align-y="4272" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="false">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Notice the difference in display between ranges and vectors:</p>
</text>
</region>
<region region-id="3589" left="150" top="4284.75" width="42" height="70.5" align-x="150.75" align-y="4296" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:eval placeholderMultiplicationStyle="default" resultMultiplicationStyle="default">
<ml:id xml:space="preserve">i</ml:id>
<result xmlns="http://schemas.mathsoft.com/math30">
<ml:matrix rows="3" cols="1">
<ml:real>0</ml:real>
<ml:real>1</ml:real>
<ml:real>2</ml:real>
</ml:matrix>
</result>
</ml:eval>
<resultFormat>
<table item-idref="38"></table>
</resultFormat>
</math>
<rendering item-idref="39"></rendering>
</region>
<region region-id="3563" left="18" top="4287.75" width="53.25" height="57.75" align-x="33" align-y="4320" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:eval placeholderMultiplicationStyle="default" resultMultiplicationStyle="default">
<ml:id xml:space="preserve">B</ml:id>
<result xmlns="http://schemas.mathsoft.com/math30">
<ml:matrix rows="3" cols="1">
<ml:real>4</ml:real>
<ml:real>5</ml:real>
<ml:real>6</ml:real>
</ml:matrix>
</result>
</ml:eval>
</math>
<rendering item-idref="40"></rendering>
</region>
<region region-id="3580" left="6" top="4356.75" width="330.75" height="71.25" align-x="6" align-y="4368" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">They behave differently as well. Vectors can be assigned to other vectors and produce vectors when used in element-wise calculations. Ranges produce ranges and imply iteration over an expression. So, while you can produce a vector and assign it to another vector, if you produce a range, you must assign it to a vector indexed expression: </p>
</text>
</region>
<region region-id="3598" left="18" top="4440.75" width="86.25" height="15.75" align-x="60.75" align-y="4452" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:define>
<ml:id xml:space="preserve">cosVec</ml:id>
<ml:apply>
<ml:id xml:space="preserve">cos</ml:id>
<ml:id xml:space="preserve">B</ml:id>
</ml:apply>
</ml:define>
</math>
<rendering item-idref="41"></rendering>
</region>
<region region-id="3599" left="150" top="4440.75" width="90.75" height="15.75" align-x="202.5" align-y="4452" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math error="This value must be a scalar or a vector." optimize="false" disable-calc="false">
<ml:define>
<ml:id xml:space="preserve">cosRange</ml:id>
<ml:apply>
<ml:id xml:space="preserve">cos</ml:id>
<ml:id xml:space="preserve">i</ml:id>
</ml:apply>
</ml:define>
</math>
<metadata>
<customValue name="Mathsoft_expected_error" type="text">must be a scalar or a vector</customValue>
</metadata>
<rendering item-idref="42"></rendering>
</region>
<region region-id="3601" left="6" top="4470.75" width="315.75" height="57" align-x="6" align-y="4482" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">The second assignment doesn't work, because iteration on the right implies iteration on the left, and you have to tell Mathcad how to assign the results of each iteration. Doing this with subscripts creates a vector.</p>
</text>
</region>
<region region-id="3607" left="18" top="4554.75" width="93" height="17.25" align-x="72.75" align-y="4566" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:define>
<ml:apply>
<ml:indexer></ml:indexer>
<ml:id xml:space="preserve">cosRange</ml:id>
<ml:id xml:space="preserve">i</ml:id>
</ml:apply>
<ml:apply>
<ml:id xml:space="preserve">cos</ml:id>
<ml:id xml:space="preserve">i</ml:id>
</ml:apply>
</ml:define>
</math>
<rendering item-idref="43"></rendering>
</region>
<region region-id="3606" left="150" top="4533.75" width="117.75" height="57.75" align-x="201.75" align-y="4566" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:eval placeholderMultiplicationStyle="default" resultMultiplicationStyle="default">
<ml:id xml:space="preserve">cosRange</ml:id>
<result xmlns="http://schemas.mathsoft.com/math30">
<ml:matrix rows="3" cols="1">
<ml:real>1</ml:real>
<ml:real>0.54030230586813976</ml:real>
<ml:real>-0.41614683654714241</ml:real>
</ml:matrix>
</result>
</ml:eval>
</math>
<rendering item-idref="44"></rendering>
</region>
<region region-id="3548" left="0" top="4602" width="6000" height="6" align-x="0" align-y="4602" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<pageBreak></pageBreak>
</region>
<region region-id="2545" left="6" top="4610.25" width="250.5" height="12" align-x="6" align-y="4620" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="calculating vectors">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Heading 2" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Linear Algebra with Vectors and Matrices</p>
</text>
<indexes>
<phrase>vector calculations</phrase>
<phrase>matrix calculations</phrase>
<phrase>calculations, vector and matrix</phrase>
</indexes>
</region>
<region region-id="2546" left="6" top="4626.75" width="342" height="57" align-x="6" align-y="4638" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Linear algebra has some simple rules for how to calculate using arrays. Generally, you can apply the same operations to arrays as you would to simple variables. There are no special instructions for calculating with vectors or matrices in Mathcad. Simply follow the rules of mathematics.</p>
</text>
</region>
<region region-id="3405" left="6" top="4700.25" width="56.25" height="12" align-x="6" align-y="4710" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="false">
<p style="Heading 2" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Examples </p>
</text>
</region>
<region region-id="2976" left="6" top="4724.25" width="153.75" height="12" align-x="21" align-y="4734" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Heading 2" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Scalar Multiplication</p>
</text>
</region>
<region region-id="2547" left="6" top="4740.75" width="341.25" height="42.75" align-x="6" align-y="4752" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">If you multiply a vector by a scalar (single number), the result is a vector in which each component of the original vector has been multiplied by the scalar. </p>
</text>
</region>
<region region-id="2922" left="18" top="4793.25" width="49.5" height="78.75" align-x="34.5" align-y="4836" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:define>
<ml:id xml:space="preserve">D</ml:id>
<ml:matrix rows="4" cols="1">
<ml:real>2</ml:real>
<ml:real>4</ml:real>
<ml:real>5</ml:real>
<ml:real>6</ml:real>
</ml:matrix>
</ml:define>
</math>
<rendering item-idref="45"></rendering>
</region>
<region region-id="2923" left="96" top="4793.25" width="66" height="78.75" align-x="117.75" align-y="4836" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:eval placeholderMultiplicationStyle="default" resultMultiplicationStyle="default">
<ml:apply>
<ml:mult style="default"></ml:mult>
<ml:real>2</ml:real>
<ml:id xml:space="preserve">D</ml:id>
</ml:apply>
<result xmlns="http://schemas.mathsoft.com/math30">
<ml:matrix rows="4" cols="1">
<ml:real>4</ml:real>
<ml:real>8</ml:real>
<ml:real>10</ml:real>
<ml:real>12</ml:real>
</ml:matrix>
</result>
</ml:eval>
</math>
<rendering item-idref="46"></rendering>
</region>
<region region-id="2979" left="6" top="4886.25" width="228" height="12" align-x="21.75" align-y="4896" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Heading 2" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">To Add or Subtract Vectors</p>
</text>
</region>
<region region-id="2550" left="6" top="4902.75" width="368.25" height="14.25" align-x="6" align-y="4914" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">You may add or subtract two vectors whenever they are the same length. </p>
</text>
</region>
<region region-id="2928" left="18" top="4943.25" width="54" height="78.75" align-x="33.75" align-y="4986" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:eval placeholderMultiplicationStyle="default" resultMultiplicationStyle="default">
<ml:id xml:space="preserve">D</ml:id>
<result xmlns="http://schemas.mathsoft.com/math30">
<ml:matrix rows="4" cols="1">
<ml:real>2</ml:real>
<ml:real>4</ml:real>
<ml:real>5</ml:real>
<ml:real>6</ml:real>
</ml:matrix>
</result>
</ml:eval>
</math>
<rendering item-idref="47"></rendering>
</region>
<region region-id="2955" left="90" top="4953.75" width="53.25" height="57.75" align-x="105" align-y="4986" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:eval placeholderMultiplicationStyle="default" resultMultiplicationStyle="default">
<ml:id xml:space="preserve">B</ml:id>
<result xmlns="http://schemas.mathsoft.com/math30">
<ml:matrix rows="3" cols="1">
<ml:real>4</ml:real>
<ml:real>5</ml:real>
<ml:real>6</ml:real>
</ml:matrix>
</result>
</ml:eval>
</math>
<rendering item-idref="48"></rendering>
</region>
<region region-id="2956" left="162" top="4943.25" width="48" height="78.75" align-x="177" align-y="4986" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:define>
<ml:id xml:space="preserve">E</ml:id>
<ml:matrix rows="4" cols="1">
<ml:real>8</ml:real>
<ml:real>7</ml:real>
<ml:real>6</ml:real>
<ml:real>5</ml:real>
</ml:matrix>
</ml:define>
</math>
<rendering item-idref="49"></rendering>
</region>
<region region-id="3612" left="6" top="5052.75" width="62.25" height="28.5" align-x="15" align-y="5064" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">This addition works:</p>
</text>
</region>
<region region-id="2964" left="78" top="5033.25" width="80.25" height="78.75" align-x="114" align-y="5076" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:eval placeholderMultiplicationStyle="default" resultMultiplicationStyle="default">
<ml:apply>
<ml:plus></ml:plus>
<ml:id xml:space="preserve">D</ml:id>
<ml:id xml:space="preserve">E</ml:id>
</ml:apply>
<result xmlns="http://schemas.mathsoft.com/math30">
<ml:matrix rows="4" cols="1">
<ml:real>10</ml:real>
<ml:real>11</ml:real>
<ml:real>11</ml:real>
<ml:real>11</ml:real>
</ml:matrix>
</result>
</ml:eval>
</math>
<rendering item-idref="50"></rendering>
</region>
<region region-id="3544" left="6" top="5112.75" width="57.75" height="28.5" align-x="15" align-y="5124" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">This addition doesn't:</p>
</text>
</region>
<region region-id="2971" left="78" top="5124.75" width="57.75" height="15.75" align-x="114.75" align-y="5136" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math error="These array dimensions do not match." optimize="false" disable-calc="false">
<ml:eval placeholderMultiplicationStyle="default" resultMultiplicationStyle="default">
<ml:apply>
<ml:plus></ml:plus>
<ml:id xml:space="preserve">D</ml:id>
<ml:id xml:space="preserve">B</ml:id>
</ml:apply>
</ml:eval>
</math>
<metadata>
<customValue name="Mathsoft_expected_error" type="text">array size mismatch</customValue>
</metadata>
<rendering item-idref="51"></rendering>
</region>
<region region-id="3049" left="0" top="5160" width="6000" height="6" align-x="0" align-y="5160" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<pageBreak></pageBreak>
</region>
<region region-id="2981" left="6" top="5168.25" width="243.75" height="12" align-x="21.75" align-y="5178" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Heading 2" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">To Add or Subtract Matrices</p>
</text>
<indexes>
<phrase>matrices, adding and subtracting</phrase>
</indexes>
</region>
<region region-id="2558" left="6" top="5184.75" width="174.75" height="14.25" align-x="6" align-y="5196" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Here are two 2 <f family="Arial" charset="0" size="10">x</f> 2 matrices, M and P:</p>
</text>
</region>
<region region-id="2982" left="18" top="5210.25" width="66" height="36.75" align-x="36.75" align-y="5232" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:define>
<ml:id xml:space="preserve">M</ml:id>
<ml:matrix rows="2" cols="2">
<ml:real>1</ml:real>
<ml:real>2</ml:real>
<ml:real>3</ml:real>
<ml:real>4</ml:real>
</ml:matrix>
</ml:define>
</math>
<rendering item-idref="52"></rendering>
</region>
<region region-id="2983" left="132" top="5210.25" width="62.25" height="36.75" align-x="147" align-y="5232" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:define>
<ml:id xml:space="preserve">P</ml:id>
<ml:matrix rows="2" cols="2">
<ml:real>5</ml:real>
<ml:real>6</ml:real>
<ml:real>7</ml:real>
<ml:real>8</ml:real>
</ml:matrix>
</ml:define>
</math>
<rendering item-idref="53"></rendering>
</region>
<region region-id="2562" left="6" top="5256.75" width="110.25" height="14.25" align-x="6" align-y="5268" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Now add them together:</p>
</text>
</region>
<region region-id="2984" left="18" top="5282.25" width="96.75" height="36.75" align-x="56.25" align-y="5304" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:eval placeholderMultiplicationStyle="default" resultMultiplicationStyle="default">
<ml:apply>
<ml:plus></ml:plus>
<ml:id xml:space="preserve">M</ml:id>
<ml:id xml:space="preserve">P</ml:id>
</ml:apply>
<result xmlns="http://schemas.mathsoft.com/math30">
<ml:matrix rows="2" cols="2">
<ml:real>6</ml:real>
<ml:real>8</ml:real>
<ml:real>10</ml:real>
<ml:real>12</ml:real>
</ml:matrix>
</result>
</ml:eval>
</math>
<rendering item-idref="54"></rendering>
</region>
<region region-id="2565" left="6" top="5328.75" width="340.5" height="42.75" align-x="6" align-y="5340" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">When you add two matrices together, the elements of the resultant matrix are the sums of the elements of the two original matrices. Just as with vectors, matrix addition is only defined for matrices of the same size. </p>
</text>
</region>
<region region-id="3027" left="6" top="5390.25" width="233.25" height="12" align-x="6" align-y="5400" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="transpose">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Heading 2" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Transpose, Inverse, Determinant</p>
</text>
<indexes>
<phrase>inverse of a matrix</phrase>
<phrase>transpose of a matrix</phrase>
<phrase>determinant of a matrix</phrase>
</indexes>
</region>
<region region-id="3464" left="6" top="5412.75" width="348.75" height="14.25" align-x="15" align-y="5424" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="false">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">You can take the transpose, inverse, and determinant of matrices in Mathcad.</p>
</text>
</region>
<region region-id="2993" left="6" top="5436.75" width="135" height="14.25" align-x="16.5" align-y="5448" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Start with a 3 <f family="Arial" charset="0" size="10">x</f> 3 matrix:</p>
</text>
</region>
<region region-id="2994" left="18" top="5463.75" width="84" height="57.75" align-x="34.5" align-y="5496" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:define>
<ml:id xml:space="preserve">X</ml:id>
<ml:matrix rows="3" cols="3">
<ml:real>5</ml:real>
<ml:real>1</ml:real>
<ml:real>2</ml:real>
<ml:real>4</ml:real>
<ml:real>3</ml:real>
<ml:real>4</ml:real>
<ml:real>86</ml:real>
<ml:real>5</ml:real>
<ml:real>2</ml:real>
</ml:matrix>
</ml:define>
</math>
<rendering item-idref="55"></rendering>
</region>
<region region-id="2996" left="6" top="5538.75" width="279" height="14.25" align-x="6" align-y="5550" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">To calculate the transpose of the matrix, type <f family="Tahoma" charset="0" size="9">
<b>X[Ctrl]1=</b>
</f>.</p>
</text>
</region>
<region region-id="2997" left="18" top="5565.75" width="95.25" height="57.75" align-x="40.5" align-y="5598" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:eval placeholderMultiplicationStyle="default" resultMultiplicationStyle="default">
<ml:apply>
<ml:transpose></ml:transpose>
<ml:id xml:space="preserve">X</ml:id>
</ml:apply>
<result xmlns="http://schemas.mathsoft.com/math30">
<ml:matrix rows="3" cols="3">
<ml:real>5</ml:real>
<ml:real>4</ml:real>
<ml:real>86</ml:real>
<ml:real>1</ml:real>
<ml:real>3</ml:real>
<ml:real>5</ml:real>
<ml:real>2</ml:real>
<ml:real>4</ml:real>
<ml:real>2</ml:real>
</ml:matrix>
</result>
</ml:eval>
</math>
<rendering item-idref="56"></rendering>
</region>
<region region-id="3001" left="6" top="5646.75" width="237" height="14.25" align-x="6" align-y="5658" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">To calculate the inverse of the matrix, type <f family="Tahoma" charset="0" size="9">
<b>X^-1=</b>
</f>.</p>
</text>
</region>
<region region-id="3007" left="18" top="5683.5" width="203.25" height="63" align-x="47.25" align-y="5718" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:eval placeholderMultiplicationStyle="default" resultMultiplicationStyle="default">
<ml:apply>
<ml:pow></ml:pow>
<ml:id xml:space="preserve">X</ml:id>
<ml:real>-1</ml:real>
</ml:apply>
<result xmlns="http://schemas.mathsoft.com/math30">
<ml:matrix rows="3" cols="3">
<ml:real>0.06666666666666668</ml:real>
<ml:real>-0.0380952380952381</ml:real>
<ml:real>0.0095238095238095247</ml:real>
<ml:real>-1.6</ml:real>
<ml:real>0.77142857142857146</ml:real>
<ml:real>0.057142857142857141</ml:real>
<ml:real>1.1333333333333333</ml:real>
<ml:real>-0.29047619047619044</ml:real>
<ml:real>-0.052380952380952382</ml:real>
</ml:matrix>
</result>
</ml:eval>
</math>
<rendering item-idref="57"></rendering>
</region>
<region region-id="3608" left="0" top="5754" width="6000" height="6" align-x="0" align-y="5754" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<pageBreak></pageBreak>
</region>
<region region-id="3053" left="6" top="5766.75" width="285.75" height="14.25" align-x="6" align-y="5778" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">To calculate the determinant of the matrix, type <f family="Tahoma" charset="0" size="9">
<b>X[Shift][|]=</b>
</f>
</p>
</text>
</region>
<region region-id="3008" left="18" top="5796.75" width="66.75" height="15.75" align-x="44.25" align-y="5808" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:eval placeholderMultiplicationStyle="default" resultMultiplicationStyle="default">
<ml:apply>
<ml:determinant></ml:determinant>
<ml:id xml:space="preserve">X</ml:id>
</ml:apply>
<result xmlns="http://schemas.mathsoft.com/math30">
<ml:real>-210.00000000000006</ml:real>
</result>
</ml:eval>
</math>
<rendering item-idref="58"></rendering>
</region>
<region region-id="3470" left="6" top="5820.75" width="337.5" height="28.5" align-x="9.75" align-y="5832" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Right-click on the |X| above, to see that it is identified as a Square Matrix Determinant rather than an Absolute Value. </p>
</text>
</region>
<region region-id="2576" left="6" top="5864.25" width="138.75" height="12" align-x="6" align-y="5874" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="functions">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Heading 2" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Array Functions</p>
</text>
<indexes>
<phrase>array functions</phrase>
</indexes>
</region>
<region region-id="2577" left="6" top="5880.75" width="341.25" height="57" align-x="6" align-y="5892" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Mathcad includes a variety of functions and operators for working with vectors and matrices. A few are listed below. For a full list, choose <b>Function</b> from the <b>Insert</b> menu and click on the Vector and Matrix category or choose operators from the <b>Matrix Toolbar</b>.</p>
</text>
</region>
<region region-id="2935" left="18" top="5946.75" width="37.5" height="15.75" align-x="36.75" align-y="5958" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:apply>
<ml:id xml:space="preserve">max</ml:id>
<ml:id xml:space="preserve">A</ml:id>
</ml:apply>
</math>
<rendering item-idref="59"></rendering>
</region>
<region region-id="2936" left="108" top="5946.75" width="207.75" height="28.5" align-x="108" align-y="5958" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Comment" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">the maximum or minimum of the elements in a vector or matrix </p>
</text>
</region>
<region region-id="2937" left="18" top="5964.75" width="34.5" height="15.75" align-x="33.75" align-y="5976" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:apply>
<ml:id xml:space="preserve">min</ml:id>
<ml:id xml:space="preserve">A</ml:id>
</ml:apply>
</math>
<rendering item-idref="60"></rendering>
</region>
<region region-id="2938" left="18" top="5994.75" width="41.25" height="15.75" align-x="40.5" align-y="6006" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:apply>
<ml:id xml:space="preserve">rows</ml:id>
<ml:id xml:space="preserve">A</ml:id>
</ml:apply>
</math>
<rendering item-idref="61"></rendering>
</region>
<region region-id="2939" left="108" top="5994" width="176.25" height="30.75" align-x="108" align-y="6006" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Comment" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">the number of rows or columns in <region region-id="3068" left="261.75" top="5994.75" width="9" height="15.75" align-x="265.5" align-y="6006" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:id xml:space="preserve">A</ml:id>
</math>
<rendering item-idref="62"></rendering>
</region>, or the length of a column vector </p>
</text>
</region>
<region region-id="2940" left="18" top="6012.75" width="36.75" height="15.75" align-x="36" align-y="6024" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:apply>
<ml:id xml:space="preserve">cols</ml:id>
<ml:id xml:space="preserve">A</ml:id>
</ml:apply>
</math>
<rendering item-idref="63"></rendering>
</region>
<region region-id="3067" left="18" top="6030.75" width="45" height="15.75" align-x="44.25" align-y="6042" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:apply>
<ml:id xml:space="preserve">length</ml:id>
<ml:id xml:space="preserve">A</ml:id>
</ml:apply>
</math>
<rendering item-idref="64"></rendering>
</region>
<region region-id="3069" left="18" top="6060.75" width="33" height="15.75" align-x="35.25" align-y="6072" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:apply>
<ml:id xml:space="preserve">sort</ml:id>
<ml:id xml:space="preserve">v</ml:id>
</ml:apply>
</math>
<rendering item-idref="65"></rendering>
</region>
<region region-id="3070" left="108" top="6060.75" width="162.75" height="14.25" align-x="108" align-y="6072" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Comment" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">sorts a vector in ascending order</p>
</text>
</region>
<region region-id="2943" left="18" top="6078.75" width="52.5" height="15.75" align-x="40.5" align-y="6090" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:apply>
<ml:id xml:space="preserve">csort</ml:id>
<ml:sequence>
<ml:id xml:space="preserve">A</ml:id>
<ml:id xml:space="preserve">n</ml:id>
</ml:sequence>
</ml:apply>
</math>
<rendering item-idref="66"></rendering>
</region>
<region region-id="2944" left="108" top="6078" width="205.5" height="30.75" align-x="108" align-y="6090" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Comment" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">rearranges the rows of a matrix <region region-id="429" left="251.25" top="6078.75" width="9" height="15.75" align-x="255" align-y="6090" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:id xml:space="preserve">A</ml:id>
</math>
<rendering item-idref="67"></rendering>
</region> so that the nth column is in ascending order</p>
</text>
</region>
<region region-id="2945" left="18" top="6120.75" width="48" height="15.75" align-x="50.25" align-y="6132" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:apply>
<ml:id xml:space="preserve">identity</ml:id>
<ml:id xml:space="preserve">n</ml:id>
</ml:apply>
</math>
<rendering item-idref="68"></rendering>
</region>
<region region-id="2946" left="108" top="6120.75" width="147.75" height="14.25" align-x="108" align-y="6132" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Comment" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">gives the n x n identity matrix</p>
</text>
</region>
<region region-id="2947" left="108" top="6144.75" width="172.5" height="28.5" align-x="108" align-y="6156" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Comment" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">combines two matrices together by placing them side by side </p>
</text>
</region>
<region region-id="2948" left="18" top="6150.75" width="69.75" height="15.75" align-x="55.5" align-y="6162" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:apply>
<ml:id xml:space="preserve">augment</ml:id>
<ml:sequence>
<ml:id xml:space="preserve">A</ml:id>
<ml:id xml:space="preserve">B</ml:id>
</ml:sequence>
</ml:apply>
</math>
<rendering item-idref="69"></rendering>
</region>
<region region-id="2949" left="18" top="6186.75" width="56.25" height="15.75" align-x="42" align-y="6198" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:apply>
<ml:id xml:space="preserve">stack</ml:id>
<ml:sequence>
<ml:id xml:space="preserve">A</ml:id>
<ml:id xml:space="preserve">B</ml:id>
</ml:sequence>
</ml:apply>
</math>
<rendering item-idref="70"></rendering>
</region>
<region region-id="2950" left="108" top="6186.75" width="176.25" height="28.5" align-x="108" align-y="6198" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Comment" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">combines two matrices by stacking A above B</p>
</text>
</region>
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<pageBreak></pageBreak>
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<text use-page-width="false" push-down="false" lock-width="true">
<p style="Heading 2" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Repeated or Element-wise Calculations with Arrays</p>
</text>
<indexes>
<phrase>batch processing</phrase>
<phrase>repeated calculations</phrase>
<phrase>array calculations</phrase>
</indexes>
</region>
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<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">There is one array operator in particular that is useful for performing calculations in a batch, called the <b>vectorize operator</b>. For example, in Mathcad, if you multiply a vector by a scalar,</p>
</text>
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<math optimize="false" disable-calc="false">
<ml:define>
<ml:id xml:space="preserve">vec</ml:id>
<ml:matrix rows="3" cols="1">
<ml:real>1</ml:real>
<ml:real>3</ml:real>
<ml:real>5</ml:real>
</ml:matrix>
</ml:define>
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<math optimize="false" disable-calc="false">
<ml:eval placeholderMultiplicationStyle="default" resultMultiplicationStyle="default">
<ml:apply>
<ml:mult style="dot"></ml:mult>
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<ml:id xml:space="preserve">vec</ml:id>
</ml:apply>
<result xmlns="http://schemas.mathsoft.com/math30">
<ml:matrix rows="3" cols="1">
<ml:real>2</ml:real>
<ml:real>6</ml:real>
<ml:real>10</ml:real>
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<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">you get another vector back, but if you multiply two vectors,</p>
</text>
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<math optimize="false" disable-calc="false">
<ml:eval placeholderMultiplicationStyle="default" resultMultiplicationStyle="default">
<ml:apply>
<ml:mult style="dot"></ml:mult>
<ml:id xml:space="preserve">vec</ml:id>
<ml:id xml:space="preserve">vec</ml:id>
</ml:apply>
<result xmlns="http://schemas.mathsoft.com/math30">
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<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">Mathcad assumes you mean the dot product. </p>
</text>
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<text use-page-width="false" push-down="false" lock-width="false">
<p style="Heading 2" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Examples </p>
</text>
</region>
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<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Suppose you wanted element-by-element multiplication, producing another vector of the same length:</p>
</text>
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<math optimize="false" disable-calc="false">
<ml:eval placeholderMultiplicationStyle="default" resultMultiplicationStyle="default">
<ml:apply>
<ml:vectorize></ml:vectorize>
<ml:parens>
<ml:apply>
<ml:mult style="dot"></ml:mult>
<ml:matrix rows="3" cols="1">
<ml:real>1</ml:real>
<ml:real>3</ml:real>
<ml:real>5</ml:real>
</ml:matrix>
<ml:matrix rows="3" cols="1">
<ml:real>1</ml:real>
<ml:real>3</ml:real>
<ml:real>5</ml:real>
</ml:matrix>
</ml:apply>
</ml:parens>
</ml:apply>
<result xmlns="http://schemas.mathsoft.com/math30">
<ml:matrix rows="3" cols="1">
<ml:real>1</ml:real>
<ml:real>9</ml:real>
<ml:real>25</ml:real>
</ml:matrix>
</result>
</ml:eval>
</math>
<rendering item-idref="74"></rendering>
</region>
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<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">This operation is accomplished with the vectorize operator. If you type <f family="Tahoma" charset="0" size="9">
<b>M*M [Spacebar][Ctrl][-][</b>
</f>
<f family="Tahoma" charset="0" size="9">
<inlineAttr font-weight="normal">=</inlineAttr>
</f>
<f family="Tahoma" charset="0" size="9">
<b>]</b>
</f>, you get the product of each element of matrix <b>M </b>
<inlineAttr font-weight="normal">with itself</inlineAttr>:</p>
</text>
</region>
<region region-id="3106" left="24" top="6650.25" width="103.5" height="36.75" align-x="69" align-y="6672" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:eval placeholderMultiplicationStyle="default" resultMultiplicationStyle="default">
<ml:apply>
<ml:vectorize></ml:vectorize>
<ml:parens>
<ml:apply>
<ml:mult style="dot"></ml:mult>
<ml:id xml:space="preserve">M</ml:id>
<ml:id xml:space="preserve">M</ml:id>
</ml:apply>
</ml:parens>
</ml:apply>
<result xmlns="http://schemas.mathsoft.com/math30">
<ml:matrix rows="2" cols="2">
<ml:real>1</ml:real>
<ml:real>4</ml:real>
<ml:real>9</ml:real>
<ml:real>16</ml:real>
</ml:matrix>
</result>
</ml:eval>
</math>
<rendering item-idref="75"></rendering>
</region>
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<math optimize="false" disable-calc="false">
<ml:eval placeholderMultiplicationStyle="default" resultMultiplicationStyle="default">
<ml:id xml:space="preserve">M</ml:id>
<result xmlns="http://schemas.mathsoft.com/math30">
<ml:matrix rows="2" cols="2">
<ml:real>1</ml:real>
<ml:real>2</ml:real>
<ml:real>3</ml:real>
<ml:real>4</ml:real>
</ml:matrix>
</result>
</ml:eval>
</math>
<rendering item-idref="76"></rendering>
</region>
<region region-id="3110" left="6" top="6702.75" width="332.25" height="28.5" align-x="6" align-y="6714" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">You can see more advanced <b>
<u>
<link href=".\..\..\Techniques\mo90qvec.xmcd" title="Links to section" popup="false">examples of the vectorize operator</link>
</u>
</b> in QuickSheets.</p>
</text>
</region>
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<pageBreak></pageBreak>
</region>
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<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">Calculations can also be performed repeatedly using range variables to step through the elements of vectors and perform the same operations:</p>
</text>
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<math optimize="false" disable-calc="false">
<ml:define>
<ml:id xml:space="preserve">i</ml:id>
<ml:range>
<ml:real>0</ml:real>
<ml:apply>
<ml:id xml:space="preserve">last</ml:id>
<ml:id xml:space="preserve">vec</ml:id>
</ml:apply>
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</math>
<rendering item-idref="77"></rendering>
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<math optimize="false" disable-calc="false">
<ml:define>
<ml:apply>
<ml:indexer></ml:indexer>
<ml:id xml:space="preserve">newvec</ml:id>
<ml:id xml:space="preserve">i</ml:id>
</ml:apply>
<ml:apply>
<ml:mult style="default"></ml:mult>
<ml:apply>
<ml:indexer></ml:indexer>
<ml:id xml:space="preserve">vec</ml:id>
<ml:id xml:space="preserve">i</ml:id>
</ml:apply>
<ml:apply>
<ml:indexer></ml:indexer>
<ml:id xml:space="preserve">vec</ml:id>
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<math optimize="false" disable-calc="false">
<ml:eval placeholderMultiplicationStyle="default" resultMultiplicationStyle="default">
<ml:id xml:space="preserve">newvec</ml:id>
<result xmlns="http://schemas.mathsoft.com/math30">
<ml:matrix rows="3" cols="1">
<ml:real>1</ml:real>
<ml:real>9</ml:real>
<ml:real>25</ml:real>
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</math>
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</region>
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<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">This construction is particularly useful for manipulating vectors that contain signals or for solving difference equations.</p>
</text>
</region>
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<text use-page-width="false" push-down="false" lock-width="false">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">To create solutions to the logistic map:</p>
</text>
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<math optimize="false" disable-calc="false">
<ml:define>
<ml:id xml:space="preserve">N</ml:id>
<ml:real>200</ml:real>
</ml:define>
</math>
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</region>
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<math optimize="false" disable-calc="false">
<ml:define>
<ml:id xml:space="preserve">i</ml:id>
<ml:range>
<ml:real>0</ml:real>
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</ml:range>
</ml:define>
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<ml:id xml:space="preserve">n</ml:id>
<ml:range>
<ml:real>0</ml:real>
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</ml:define>
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<math optimize="false" disable-calc="false">
<ml:define>
<ml:apply>
<ml:indexer></ml:indexer>
<ml:id xml:space="preserve">μ</ml:id>
<ml:id xml:space="preserve">i</ml:id>
</ml:apply>
<ml:apply>
<ml:mult style="default"></ml:mult>
<ml:real>4</ml:real>
<ml:apply>
<ml:div></ml:div>
<ml:id xml:space="preserve">i</ml:id>
<ml:id xml:space="preserve">N</ml:id>
</ml:apply>
</ml:apply>
</ml:define>
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<text use-page-width="false" push-down="false" lock-width="false">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">initialize the matrix:</p>
</text>
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<math optimize="false" disable-calc="false">
<ml:define>
<ml:apply>
<ml:indexer></ml:indexer>
<ml:id xml:space="preserve">X</ml:id>
<ml:sequence>
<ml:real>0</ml:real>
<ml:id xml:space="preserve">i</ml:id>
</ml:sequence>
</ml:apply>
<ml:apply>
<ml:id xml:space="preserve">rnd</ml:id>
<ml:real>1</ml:real>
</ml:apply>
</ml:define>
</math>
<rendering item-idref="84"></rendering>
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<math optimize="false" disable-calc="false">
<ml:define>
<ml:apply>
<ml:indexer></ml:indexer>
<ml:id xml:space="preserve">X</ml:id>
<ml:sequence>
<ml:apply>
<ml:plus></ml:plus>
<ml:id xml:space="preserve">n</ml:id>
<ml:real>1</ml:real>
</ml:apply>
<ml:id xml:space="preserve">i</ml:id>
</ml:sequence>
</ml:apply>
<ml:apply>
<ml:mult style="default"></ml:mult>
<ml:apply>
<ml:mult style="default"></ml:mult>
<ml:apply>
<ml:indexer></ml:indexer>
<ml:id xml:space="preserve">μ</ml:id>
<ml:id xml:space="preserve">i</ml:id>
</ml:apply>
<ml:apply>
<ml:indexer></ml:indexer>
<ml:id xml:space="preserve">X</ml:id>
<ml:sequence>
<ml:id xml:space="preserve">n</ml:id>
<ml:id xml:space="preserve">i</ml:id>
</ml:sequence>
</ml:apply>
</ml:apply>
<ml:parens>
<ml:apply>
<ml:minus></ml:minus>
<ml:real>1</ml:real>
<ml:apply>
<ml:indexer></ml:indexer>
<ml:id xml:space="preserve">X</ml:id>
<ml:sequence>
<ml:id xml:space="preserve">n</ml:id>
<ml:id xml:space="preserve">i</ml:id>
</ml:sequence>
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</ml:apply>
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</ml:apply>
</ml:define>
</math>
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<math optimize="false" disable-calc="false">
<ml:define>
<ml:id xml:space="preserve">n</ml:id>
<ml:range>
<ml:apply>
<ml:div></ml:div>
<ml:id xml:space="preserve">N</ml:id>
<ml:real>2</ml:real>
</ml:apply>
<ml:id xml:space="preserve">N</ml:id>
</ml:range>
</ml:define>
</math>
<rendering item-idref="86"></rendering>
</region>
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<plot disable-calc="false" item-idref="87"></plot>
<rendering item-idref="88"></rendering>
</region>
<region region-id="3370" left="6" top="7158.75" width="327" height="42.75" align-x="18" align-y="7170" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">More detailed examples are available on <b>
<u>
<link href=".\..\..\Solving\a080seed.xmcd" title="Links to section" popup="false">seeded iteration</link>
</u>
</b> using range variables and arrays, as well as on <b>
<u>
<link href=".\..\..\Solving\s020ufrt.xmcd" title="Links to section" popup="false">parametric sets of solutions</link>
</u>
</b> to equations.</p>
</text>
</region>
<region region-id="3353" left="6" top="7218.75" width="111" height="14.25" align-x="6" align-y="7230" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">To convolve two signals:</p>
</text>
</region>
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<math optimize="false" disable-calc="false">
<ml:define>
<ml:id xml:space="preserve">N</ml:id>
<ml:real>20</ml:real>
</ml:define>
</math>
<rendering item-idref="89"></rendering>
</region>
<region region-id="3356" left="102" top="7240.5" width="37.5" height="33.75" align-x="117" align-y="7260" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:define>
<ml:id xml:space="preserve">T</ml:id>
<ml:apply>
<ml:div></ml:div>
<ml:real>1</ml:real>
<ml:id xml:space="preserve">N</ml:id>
</ml:apply>
</ml:define>
</math>
<rendering item-idref="90"></rendering>
</region>
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<math optimize="false" disable-calc="false">
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<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">For a more detailed example of signals handled with vectors in Mathcad, see the <link href=".\..\..\Applications\f060.xmcd" title="Links to section" popup="false">
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