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  <question type="description">
    <name>
      <text>9.2.1.1 Two-dimensional coordinate frames</text>
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      <text><![CDATA[<h4>Two-dimensional coordinate frames<br></h4><p>The figure below shows a two-dimensional coordinate frame. Any point <span class="math inline"><em>P</em></span> in the \(xy\) plane can be defined in terms of its \(x\) and \(y\) coordinates.</p>
<figure class="HELM_figure">
<img src="@@PLUGINFILE@@/9-2-1PointInxy.svg" alt="A set of x-y axes with a point P marked on it at position P" class="img-responsive atto_image_button_text-bottom" width="215" height="164">
</figure>

<p>A unit vector pointing in the positive direction of the \(x\)-axis is denoted by \(\underline{i}\). (Note that it is common practice to write this particular unit vector without the hat \(\hat{\ }\).) It follows that any vector in the direction of the \(x\)-axis will be a multiple of \(\underline{ i}\). The figure below shows vectors \(\underline{ i}\), \(2\underline{ i}\), \(5\underline{ i}\) and \(-3\underline{ i}\). In general a vector of length \(\ell\) in the direction of the \(x\)-axis can be written \(\ell\underline{ i}\).</p>
<figure class="HELM_figure">
<img src="@@PLUGINFILE@@/9-2-1xVectors.svg" alt="4 vectors of various lengths pointing in the x direction." class="img-responsive atto_image_button_text-bottom" width="275" height="164">
</figure>

<p>Similarly, a unit vector pointing in the positive \(y\)-axis is denoted by \(\underline{ j}\). So any vector in the direction of the \(y\)-axis will be a multiple of \(\underline{ j}\). The figure below shows \(\underline{ j}\), \(4\underline{ j}\) and \(-2\underline{ j}\). In general a vector of length \(\ell\) in the direction of the \(y\)-axis can be written \(\ell\underline{ j}\).</p>
<figure class="HELM_figure">
<img src="@@PLUGINFILE@@/9-2-1yVectors.svg" alt="Three vectors in the y-direction of different lengths." class="img-responsive atto_image_button_text-bottom" width="203" height="191"></p><p style="text-align: left;">
</figure>

<div class="HELM_keypoint">
<h4>Key Point<br></h4>
<p>\(\underline{i}\) represents a unit vector in the direction of the positive \(x\)-axis</p>
<p>\(\underline{j}\) represents a unit vector in the direction of the positive \(y\)-axis</p>
</div>
<hr>
<h4 class="HELM_example">Example<br></h4>
<p>Draw the vectors \(5\underline{ i}\) and \(4\underline{ j}\). Use your diagram and the triangle law of addition to add these two vectors together. First draw the vectors \(5\underline{ i}\) and \(4\underline{ j}\). Then, by translating the vectors so that they lie head to tail, find the vector sum \(5\underline{ i} + 4\underline{ j}\).</p>
<h4 class="HELM_solution">Solution</h4>
<figure class="HELM_figure">
<img src="@@PLUGINFILE@@/9-2-1Ex1.svg" alt="On the left, the two vectors 4i and 5j are drawn, starting from the same point.  On the right the are drawn with the base of 4i at the head of 5j and their sum is drawn to complete the triangle" class="img-responsive atto_image_button_text-bottom" width="564" height="187">
</figure>
<hr>
<p>We now generalise the situation in the Example above. Consider the figure below.</p>
<figure class="HELM_figure">
<img src="@@PLUGINFILE@@/9-2-1TriangleLawAdd.svg" alt="A right angle triangle ABC with sides ai, bj and hypotenuse r." class="img-responsive atto_image_button_text-bottom" width="339" height="247">
</figure>

<p>It shows a vector \(\underline{ r}=\overrightarrow{AB}\). We can regard \(\underline{ r}\) as being the resultant of the two vectors \(\overrightarrow{AC}=a\underline{ i}\) and \(\overrightarrow{CB}=b\underline{ j}\). From the triangle law of vector addition we have \[\overrightarrow{AB} =  \overrightarrow{AC}+\overrightarrow{CB} =  a\underline{i} + b\underline{j}\]</p>
<p>We conclude that any vector in the \(xy\) plane can be expressed in the form \(\underline{ r}=
a\underline{ i} + b\underline{ j}\). The numbers \(a\) and \(b\) are called the <b>components</b> of \(\underline{r}\) in the \(x\) and \(y\) directions. Sometimes, for emphasis, we will use \(a_{x}\) and \(a_{y}\) instead of \(a\) and \(b\) to denote the components in the \(x\)- and \(y\)-directions respectively. In that case we would write \(\underline{ r}=
a_x\underline{ i} + a_y\underline{ j}\).</p>]]></text>
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