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<quiz>
  <question type="stack">
    <name>
      <text>9.5.4.1 Vectors in a plane, Exercise [alt]</text>
    </name>
    <questiontext format="html">
      <text><![CDATA[<p class="HELM_exercise">Exercise</p>
<h3>The vector equation of a plane</h3>
<p>Consider the plane shown in the figure below.</p>

<figure class="HELM_figure">
<img src="@@PLUGINFILE@@/fig4vecEqPlane.svg" alt="A diagram of a 2D plane embedded in 3D vector space. Two points A and P on the plane are indicated and the vectors OA and OP, denoted by a and r, respectively, connecting the origin to each of them are sketched. Also indicated is a vector n which is normal to the plane." class="img-responsive atto_image_button_text-bottom" width="485" height="280">
</figure>

<p>Suppose that \(A\) is a fixed point in the plane and has position vector \(\underline{ a}\). Suppose that \(P\) is any other arbitrary point in the plane with position vector \(\underline{ r}\). Clearly the vector \(\overrightarrow{AP}\) lies in the plane.</p>

<ol class="HELM_parts">
    <li>
        <p>Referring to the figure, write the vector \(\overrightarrow{AP}\) in terms of \(\underline{a}\) and \(\underline{r}\). Enter the vectors \(\underline{a}\) and \(\underline{r}\) as <code>a</code> and <code>r</code>, respectively, ignoring the underline.</p>
        <p>\(\overrightarrow{AP} =\) [[input:ans1]] [[validation:ans1]]</p>
[[feedback:prt1]]
    </li>

    <li>
 <p>Also shown in the figure is a vector which is perpendicular to the plane and denoted by \(\underline{n}\).</p>
        <p>What relationship exists between \(\underline{n}\) and the vectors \(\underline{a}\) and \(\underline{r}\)?</p>
        <p>Hint: think about the scalar product of \(\underline{n}\) with \(\overrightarrow{AP}\). The scalar product \(\underline{u}\cdot\underline{v}\) should be entered as <code>u*v</code>.</p>
        <p>[[input:ans2LHS]] [[validation:ans2LHS]] \(=\) [[input:ans2RHS]] [[validation:ans2RHS]]</p>
[[feedback:prt2]]
    </li>
</ol>]]></text>
<file name="fig4vecEqPlane.svg" path="/" encoding="base64">[encoded data, 25072 characters]</file>
    </questiontext>
    <generalfeedback format="html">
      <text><![CDATA[<ol class="HELM_parts">
<li><p>An expression for \(\overrightarrow{AP}\) in terms of \(\underline{a}\) and \(\underline{r}\) is \(\overrightarrow{AP} = \underline{r} - \underline{a}\).</p></li>

<li><p>Because \(\overrightarrow{AP}\) and \(\underline{ n}\) are perpendicular their scalar product must equal zero, that is \((\underline{r}-\underline{a}) \cdot \underline{n}=0\) so that \(\underline{ r} \cdot \underline{n}=\underline{a} \cdot \underline{n}\).</p>]]></text>
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      <text>2020120600</text>
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      <text><![CDATA[stackvectortex(ex):= block(sconcat("{ \\underline{", tex1(first(args(ex))), "}}"));
ta1: r-a;
wrongans: a-r;
ta2LHS: (r-a)*n;
ta2LHSrev: n*(r-a);
ta2RHS: 0;
ta2LHSalt: r*n;
ta2RHSalt: a*n;]]></text>
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      <text>ta1: r-a;
ta2LHS: (r-a)*n;
ta2RHS: 0;</text>
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      <text><![CDATA[<span style="font-size: 1.5em; color:orange;"><i class="fa fa-adjust"></i></span> Your answer is partially correct.]]></text>
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    <prtincorrect format="html">
      <text><![CDATA[<span style="font-size: 1.5em; color:red;"><i class="fa fa-times"></i></span> Incorrect answer.]]></text>
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    <decimals>.</decimals>
    <scientificnotation>*10</scientificnotation>
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