<?xml version="1.0" encoding="UTF-8"?>
<quiz>
  <question type="stack">
    <name>
      <text>Cross-product-non-cancellation.</text>
    </name>
    <questiontext format="html">
      <text><![CDATA[<p>If \(\mathbf{a} \times \mathbf{b} = \mathbf{a} \times\mathbf{c}\) we cannot "cancel" \(\mathbf{a}\) to conclude that \(\mathbf{b} = \mathbf{c}\).</p>
<p>Instead, using the laws of vector products, we have that \(\mathbf{a} \times \mathbf{b} = \mathbf{a} \times\mathbf{c}\) is equivalent to \[\mathbf{a} \times ( \mathbf{b} - \mathbf{c}) = \mathbf{0}.\]</p>
<p>We can only infer from this that one of three possiblities is true -- which of these are the three possibilities?</p>
<p>[[input:ans1]] [[validation:ans1]]</p>]]></text>
    </questiontext>
    <generalfeedback format="html">
      <text><![CDATA[<p>A vector product can be \(\mathbf{0}\) if one of the vectors in the product is \(\mathbf{0}\), or if the two vectors are parallel.</p><p>So the three possibilities that make \(\mathbf{a} \times ( \mathbf{b} - \mathbf{c}) = \mathbf{0}\) are:</p>
<p><ul><li>\(\mathbf{a} = \mathbf{0}\);&nbsp;</li><li>\(\mathbf{b} - \mathbf{c} = \mathbf{0}\), i.e. \(\mathbf{b} = \mathbf{c}\); or</li><li>\(\mathbf{b} - \mathbf{c}\) is parallel to \(\mathbf{a}\).</li></ul></p>]]></text>
    </generalfeedback>
    <defaultgrade>1</defaultgrade>
    <penalty>0.1</penalty>
    <hidden>0</hidden>
    <idnumber/>
    <stackversion>
      <text>2024092500</text>
    </stackversion>
    <questionvariables>
      <text><![CDATA[corbase:["\\(\\mathbf{a}=0\\)", "\\(\\mathbf{b}=\\mathbf{c}\\)", "\\(\\mathbf{b}-\\mathbf{c}\\) is parallel to \\(\\mathbf{a}\\)"];
wrongbase:["\\(\\mathbf{b}=0\\)", "\\(\\mathbf{b}-\\mathbf{c}\\) is perpendicular to \\(\\mathbf{a}\\)"]; 
mcq:multiselqnalpha(corbase, 3, wrongbase, 2, ""); 
/* This final argument being "" is needed because we mix text and LaTeX. */

ta1:mcq[1];
version:mcq[2];
corr1:mcq_correct(ta1);
incorr1:mcq_incorrect(ta1);]]></text>
    </questionvariables>
    <specificfeedback format="html">
      <text>[[feedback:prt1]]</text>
    </specificfeedback>
    <questionnote format="moodle_auto_format">
      <text>{#version#}</text>
    </questionnote>
    <questiondescription format="moodle_auto_format">
      <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>
    </questiondescription>
    <questionsimplify>1</questionsimplify>
    <assumepositive>0</assumepositive>
    <assumereal>0</assumereal>
    <prtcorrect format="html">
      <text><![CDATA[<span style="font-size: 1.5em; color:green;"><i class="fa fa-check"></i></span> Correct answer, well done.]]></text>
    </prtcorrect>
    <prtpartiallycorrect format="html">
      <text><![CDATA[<span style="font-size: 1.5em; color:orange;"><i class="fa fa-adjust"></i></span> Your answer is partially correct.]]></text>
    </prtpartiallycorrect>
    <prtincorrect format="html">
      <text><![CDATA[<span style="font-size: 1.5em; color:red;"><i class="fa fa-times"></i></span> Incorrect answer.]]></text>
    </prtincorrect>
    <decimals>.</decimals>
    <scientificnotation>*10</scientificnotation>
    <multiplicationsign>none</multiplicationsign>
    <sqrtsign>1</sqrtsign>
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    <logicsymbol>lang</logicsymbol>
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    <isbroken>0</isbroken>
    <variantsselectionseed/>
    <input>
      <name>ans1</name>
      <type>checkbox</type>
      <tans>ta1</tans>
      <boxsize>15</boxsize>
      <strictsyntax>1</strictsyntax>
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      <syntaxattribute>0</syntaxattribute>
      <forbidwords/>
      <allowwords/>
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      <options/>
    </input>
    <prt>
      <name>prt1</name>
      <value>1.0000000</value>
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      <feedbackstyle>1</feedbackstyle>
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        <text/>
      </feedbackvariables>
      <node>
        <name>0</name>
        <description/>
        <answertest>AlgEquiv</answertest>
        <sans>setify(ans1)</sans>
        <tans>setify(corr1)</tans>
        <testoptions/>
        <quiet>1</quiet>
        <truescoremode>=</truescoremode>
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          <text/>
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        <falsefeedback format="html">
          <text/>
        </falsefeedback>
      </node>
      <node>
        <name>1</name>
        <description/>
        <answertest>AlgEquiv</answertest>
        <sans>subsetp(setify(ans1),setify(corr1))</sans>
        <tans>true</tans>
        <testoptions/>
        <quiet>0</quiet>
        <truescoremode>+</truescoremode>
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        <truenextnode>-1</truenextnode>
        <trueanswernote>prt1-2-T</trueanswernote>
        <truefeedback format="html">
          <text><![CDATA[<p>You have correctly identified one of the correct answers, but missed \({@sequenceify(setdifference(setify(corr1),setify(ans1)))@}\).<br></p>]]></text>
        </truefeedback>
        <falsescoremode>-</falsescoremode>
        <falsescore>0</falsescore>
        <falsepenalty/>
        <falsenextnode>2</falsenextnode>
        <falseanswernote>prt1-2-F</falseanswernote>
        <falsefeedback format="html">
          <text/>
        </falsefeedback>
      </node>
      <node>
        <name>2</name>
        <description/>
        <answertest>CasEqual</answertest>
        <sans>cardinality(ev(intersection(setify(incorr1),setify(ans1)),simp))</sans>
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          <text/>
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        <falsepenalty/>
        <falsenextnode>-1</falsenextnode>
        <falseanswernote>prt1-3-F</falseanswernote>
        <falsefeedback format="html">
          <text><![CDATA[<p>You incorrectly selected {@sequenceify(setdifference(setify(ans1),setify(corr1)))@}</p>]]></text>
        </falsefeedback>
      </node>
    </prt>
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    <qtest>
      <testcase>1</testcase>
      <description/>
      <testinput>
        <name>ans1</name>
        <value>[incorr1[1]]</value>
      </testinput>
      <expected>
        <name>prt1</name>
        <expectedscore>0.0000000</expectedscore>
        <expectedpenalty>0.1000000</expectedpenalty>
        <expectedanswernote>prt1-3-F</expectedanswernote>
      </expected>
    </qtest>
    <qtest>
      <testcase>2</testcase>
      <description/>
      <testinput>
        <name>ans1</name>
        <value>corr1</value>
      </testinput>
      <expected>
        <name>prt1</name>
        <expectedscore>1.0000000</expectedscore>
        <expectedpenalty>0.0000000</expectedpenalty>
        <expectedanswernote>prt1-0-T</expectedanswernote>
      </expected>
    </qtest>
    <qtest>
      <testcase>3</testcase>
      <description/>
      <testinput>
        <name>ans1</name>
        <value>[corr1[1],incorr1[1]]</value>
      </testinput>
      <expected>
        <name>prt1</name>
        <expectedscore>0.0000000</expectedscore>
        <expectedpenalty>0.1000000</expectedpenalty>
        <expectedanswernote>prt1-3-F</expectedanswernote>
      </expected>
    </qtest>
    <qtest>
      <testcase>4</testcase>
      <description/>
      <testinput>
        <name>ans1</name>
        <value>[corr1[1]]</value>
      </testinput>
      <expected>
        <name>prt1</name>
        <expectedscore>0.5000000</expectedscore>
        <expectedpenalty/>
        <expectedanswernote>prt1-2-T</expectedanswernote>
      </expected>
    </qtest>
    <tags>
      <tag><text>multiselqnalpha</text>
</tag>
    </tags>
  </question>
</quiz>
