<?xml version="1.0" encoding="UTF-8"?>
<quiz>
  <question type="stack">
    <name>
      <text>Dot product formula proof comprehension</text>
    </name>
    <questiontext format="html">
      <text><![CDATA[<p>Consider each line in the proof below. Select from the following list the corresponding dot product rule number which justifies each step.</p>
<p><b>Dot product rules</b>: Let \(\mathbf{u}\), \(\mathbf{v}\) and \(\mathbf{w}\) be vectors in \(\mathbb{R}^n\) and let \(c\) be a scalar.
<ol>
<li>\(\mathbf{u}\cdot\mathbf{v} = \mathbf{v}\cdot\mathbf{u}\).</li>
<li>\(\mathbf{u}\cdot(\mathbf{v}+\mathbf{w}) = \mathbf{u}\cdot\mathbf{v} + \mathbf{u}\cdot\mathbf{w}\).</li>
<li>\((c \mathbf{u}) \cdot \mathbf{v} = c(\mathbf{u} \cdot \mathbf{v})\).</li>
<li>\(\mathbf{u} \cdot \mathbf{u} \geq 0\) and \(\mathbf{u} \cdot \mathbf{u} = 0\) if and only if \(\mathbf{u} = 0\).</li>
<li>\(\mathbf{u}\cdot\mathbf{u} = ||\mathbf{u}||^2\).</li>
</ol>
</p>
<hr/>
<p>
<b>Theorem</b>:  \( \mathbf{u}\cdot\mathbf{v} = ||\mathbf{u}||\,||\mathbf{v}|| \cos(\theta)\)
</p>
<div class="proof">
<p>
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</p>
<div class="proof-line">
 <div class="proof-num">1.</div>
 <div class="proof-step">Applying the [[input:ans1]] [[validation:ans1]] to the above triangle we have </div>
</div>
<div class="proof-line">
 <div class="proof-num">2.</div>
 <div class="proof-step">
 \( ||\mathbf{u}-\mathbf{v}||^2 = ||\mathbf{u}||^2 + ||\mathbf{v}||^2 - 2||\mathbf{u}||\,||\mathbf{v}|| \cos(\theta)  \)
</div>
</div>
<div class="proof-line">
 <div class="proof-num">3.</div>
 <div class="proof-step">Expanding the left-hand side we obtain 
<table>
<tr>
 <td>\( ||\mathbf{u}-\mathbf{v}||^2 \)</td>
 <td> (<i>Using rule number:<i>)</td>
</tr>
<tr>
 <td>\( =(\mathbf{u}-\mathbf{v})\cdot (\mathbf{u}-\mathbf{v})\)</td>
 <td>[[input:ans31]] [[validation:ans31]]</td>
</tr>
<tr>
 <td>\( =(\mathbf{u}-\mathbf{v})\cdot \mathbf{u}-(\mathbf{u}-\mathbf{v})\cdot\mathbf{v}\)</td>
 <td>[[input:ans32]] [[validation:ans32]] </td>
</tr>
<tr>
 <td>\( =\mathbf{u}\cdot (\mathbf{u}-\mathbf{v}) -\mathbf{v}\cdot(\mathbf{u}-\mathbf{v})\)</td>
 <td>[[input:ans33]] [[validation:ans33]] </td>
</tr>
<tr>
 <td>\( =\mathbf{u}\cdot \mathbf{u}-\mathbf{u}\cdot\mathbf{v} -\mathbf{v}\cdot\mathbf{u}+\mathbf{v}\cdot\mathbf{v}\)</td>
 <td>[[input:ans34]] [[validation:ans34]]</td>
</tr>
<tr>
 <td>\( =\mathbf{u}\cdot \mathbf{u}-\mathbf{u}\cdot\mathbf{v}-\mathbf{u}\cdot\mathbf{v}+\mathbf{v}\cdot\mathbf{v}\)</td>
 <td>[[input:ans35]] [[validation:ans35]]</td>
</tr>
<tr>
 <td>\( = ||\mathbf{u}||^2 - 2\mathbf{u} \cdot \mathbf{v} + ||\mathbf{v}||^2\)  
</td>
 <td>[[input:ans36]] [[validation:ans36]] </td>
</tr>
</table></div>
</div>
<div class="proof-line">
 <div class="proof-num">4.</div>
 <div class="proof-step">So the equation  in step 2 becomes
\[ ||\mathbf{u}||^2 - 2\mathbf{u} \cdot \mathbf{v} + ||\mathbf{v}||^2 = ||\mathbf{u}||^2 + ||\mathbf{v}||^2 - 2||\mathbf{u}||\,||\mathbf{v}|| \cos(\theta).  \]</div>
</div>
<div class="proof-line">
 <div class="proof-num">5.</div>
 <div class="proof-step">After simplification this leaves us with \( \mathbf{u}\cdot\mathbf{v} = ||\mathbf{u}||\,||\mathbf{v}|| \cos(\theta)\).
</div>
</div>
</div>]]></text>
    </questiontext>
    <generalfeedback format="moodle_auto_format">
      <text><![CDATA[<p>In step 1 of the proof we apply the law of cosines.</p>]]></text>
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      <text>2023010400</text>
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      <text><![CDATA[tal1:[[a, false, "Pythagorean theorem"], [b, false, "triangle inequality"], [c, false, "law of sines"], [d, true, "law of cosines"], [e, false, "Skew's distance"]];
ta1:d;
ta31:5;
ta32:2;
ta33:1;
ta34:2;
ta35:1;
ta36:5;]]></text>
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    <specificfeedback format="html">
      <text><![CDATA[<div class="proof-line">
 <div class="proof-num">1.</div>
 <div class="proof-step"> [[feedback:prt1]] </div>
</div>
<div class="proof-line">
 <div class="proof-num">2.</div>
 <div class="proof-step"> \( ||\mathbf{u}-\mathbf{v}||^2 = ||\mathbf{u}||^2 + ||\mathbf{v}||^2 - 2||\mathbf{u}||\,||\mathbf{v}|| \cos(\theta)  \)
</div>
</div>
<div class="proof-line">
 <div class="proof-num">3.</div>
 <div class="proof-step">Expanding the left-hand side we obtain 
<table>
<tr>
 <td>\( ||\mathbf{u}-\mathbf{v}||^2 \)</td>
 <td> (<i>Using rule number:<i>)</td>
</tr>
<tr>
 <td>\( =(\mathbf{u}-\mathbf{v})\cdot (\mathbf{u}-\mathbf{v})\)</td>
 <td> [[feedback:prt31]]</td>
</tr>
<tr>
 <td>\( =(\mathbf{u}-\mathbf{v})\cdot \mathbf{u}-(\mathbf{u}-\mathbf{v})\cdot\mathbf{v}\)</td>
 <td> [[feedback:prt32]]</td>
</tr>
<tr>
 <td>\( =\mathbf{u}\cdot (\mathbf{u}-\mathbf{v}) -\mathbf{v}\cdot(\mathbf{u}-\mathbf{v})\)</td>
 <td>[[feedback:prt33]]</td>
</tr>
<tr>
 <td>\( =\mathbf{u}\cdot \mathbf{u}-\mathbf{u}\cdot\mathbf{v} -\mathbf{v}\cdot\mathbf{u}+\mathbf{v}\cdot\mathbf{v}\)</td>
 <td>[[feedback:prt34]]</td>
</tr>
<tr>
 <td>\( =\mathbf{u}\cdot \mathbf{u}-\mathbf{u}\cdot\mathbf{v}-\mathbf{u}\cdot\mathbf{v}+\mathbf{v}\cdot\mathbf{v}\)</td>
 <td>[[feedback:prt35]]</td>
</tr>
<tr>
 <td>\( = ||\mathbf{u}||^2 - 2\mathbf{u} \cdot \mathbf{v} + ||\mathbf{v}||^2\)  
</td>
 <td>[[feedback:prt36]]</td>
</tr>
</table></div>
</div>
<div class="proof-line">
 <div class="proof-num">4.</div>
 <div class="proof-step">So the equation  in step 2 becomes
\[ ||\mathbf{u}||^2 - 2\mathbf{u} \cdot \mathbf{v} + ||\mathbf{v}||^2 = ||\mathbf{u}||^2 + ||\mathbf{v}||^2 - 2||\mathbf{u}||\,||\mathbf{v}|| \cos(\theta).  \]</div>
</div>
<div class="proof-line">
 <div class="proof-num">5.</div>
 <div class="proof-step">After simplification this leaves us with \( \mathbf{u}\cdot\mathbf{v} = ||\mathbf{u}||\,||\mathbf{v}|| \cos(\theta)\).
</div>
</div>
</div>]]></text>
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      <text/>
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      <text>This question is distributed as part of the STACK source code as an example. This is licenced as Creative Commons Attribution-ShareAlike 4.0 International License.</text>
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        <testoptions/>
        <quiet>1</quiet>
        <truescoremode>=</truescoremode>
        <truescore>1</truescore>
        <truepenalty/>
        <truenextnode>-1</truenextnode>
        <trueanswernote>prt35-1-T</trueanswernote>
        <truefeedback format="html">
          <text/>
        </truefeedback>
        <falsescoremode>=</falsescoremode>
        <falsescore>0</falsescore>
        <falsepenalty/>
        <falsenextnode>-1</falsenextnode>
        <falseanswernote>prt35-1-F</falseanswernote>
        <falsefeedback format="html">
          <text/>
        </falsefeedback>
      </node>
    </prt>
    <prt>
      <name>prt36</name>
      <value>1.0000000</value>
      <autosimplify>1</autosimplify>
      <feedbackstyle>2</feedbackstyle>
      <feedbackvariables>
        <text/>
      </feedbackvariables>
      <node>
        <name>0</name>
        <description/>
        <answertest>AlgEquiv</answertest>
        <sans>ans36</sans>
        <tans>ta36</tans>
        <testoptions/>
        <quiet>1</quiet>
        <truescoremode>=</truescoremode>
        <truescore>1</truescore>
        <truepenalty/>
        <truenextnode>-1</truenextnode>
        <trueanswernote>prt36-1-T</trueanswernote>
        <truefeedback format="html">
          <text/>
        </truefeedback>
        <falsescoremode>=</falsescoremode>
        <falsescore>0</falsescore>
        <falsepenalty/>
        <falsenextnode>-1</falsenextnode>
        <falseanswernote>prt36-1-F</falseanswernote>
        <falsefeedback format="html">
          <text/>
        </falsefeedback>
      </node>
    </prt>
    <qtest>
      <testcase>1</testcase>
      <description/>
      <testinput>
        <name>ans1</name>
        <value>ta1</value>
      </testinput>
      <testinput>
        <name>ans31</name>
        <value>ta31</value>
      </testinput>
      <testinput>
        <name>ans32</name>
        <value>ta32</value>
      </testinput>
      <testinput>
        <name>ans33</name>
        <value>ta33</value>
      </testinput>
      <testinput>
        <name>ans34</name>
        <value>ta34</value>
      </testinput>
      <testinput>
        <name>ans35</name>
        <value>ta35</value>
      </testinput>
      <testinput>
        <name>ans36</name>
        <value>ta36</value>
      </testinput>
      <expected>
        <name>prt1</name>
        <expectedscore>1.0000000</expectedscore>
        <expectedpenalty>0.0000000</expectedpenalty>
        <expectedanswernote>prt1-1-T</expectedanswernote>
      </expected>
      <expected>
        <name>prt31</name>
        <expectedscore>1.0000000</expectedscore>
        <expectedpenalty>0.0000000</expectedpenalty>
        <expectedanswernote>prt31-1-T</expectedanswernote>
      </expected>
      <expected>
        <name>prt32</name>
        <expectedscore>1.0000000</expectedscore>
        <expectedpenalty>0.0000000</expectedpenalty>
        <expectedanswernote>prt32-1-T</expectedanswernote>
      </expected>
      <expected>
        <name>prt33</name>
        <expectedscore>1.0000000</expectedscore>
        <expectedpenalty>0.0000000</expectedpenalty>
        <expectedanswernote>prt33-1-T</expectedanswernote>
      </expected>
      <expected>
        <name>prt34</name>
        <expectedscore>1.0000000</expectedscore>
        <expectedpenalty>0.0000000</expectedpenalty>
        <expectedanswernote>prt34-1-T</expectedanswernote>
      </expected>
      <expected>
        <name>prt35</name>
        <expectedscore>1.0000000</expectedscore>
        <expectedpenalty>0.0000000</expectedpenalty>
        <expectedanswernote>prt35-1-T</expectedanswernote>
      </expected>
      <expected>
        <name>prt36</name>
        <expectedscore>1.0000000</expectedscore>
        <expectedpenalty>0.0000000</expectedpenalty>
        <expectedanswernote>prt36-1-T</expectedanswernote>
      </expected>
    </qtest>
  </question>
</quiz>
