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      <text>9.1.2.1 The mathematical description of vector quantities</text>
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      <text><![CDATA[<h4>The mathematical description of vector quantities<br></h4><p>Because a vector has a direction as well as a magnitude we can represent a vector by drawing a line. The length of the line represents the <span><strong>magnitude</strong></span> of the vector given some appropriate scale, and the direction of the line represents the <span><strong>direction</strong></span> of the vector. We call this representation a <span><strong>directed line segment</strong></span>. For example, the figure below shows a vector which represents a velocity of \(3\ \mathrm{m}\,\mathrm{s}^{-1}\) north-west. Note that the arrow on the vector indicates the direction required.</p>
<figure class="HELM_figure">
<img src="@@PLUGINFILE@@/9-1-1ScaleForVector.svg" alt="An arrow represents a vector and a scale shows what length is used to represent 1m/s." class="img-responsive atto_image_button_text-bottom" width="272" height="77">
<figcaption>A vector quantity can be represented by drawing a line</figcaption>
</figure>
<p>More generally, the below figure shows an arbitrary vector quantity.</p>
<figure class="HELM_figure">
<img src="@@PLUGINFILE@@/9-1-1RepresentingVectors.svg" alt="Three identical arrows are labelled with the different ways of denoting vectors." class="img-responsive atto_image_button_text-bottom" width="537" height="97">
<figcaption>Representation of vectors</figcaption>
</figure>
<p>It is important when writing vectors to distinguish them from scalars. Various notations are used. In the figure we emphasise that we are dealing with the vector from \(A\) to \(B\) by using an <span><strong>arrow</strong></span> and writing \(\overrightarrow{AB}\). Often, in textbooks, vectors are indicated by using a <span><strong>bold typeface</strong></span> such as \(\boldsymbol{a}\). It is difficult when handwriting to reproduce the bold face and so it is conventional to <span><strong>underline</strong></span> vector quantities and write \(\underline{a}\) instead. So \(\overrightarrow{AB}\) and \(\underline{a}\) represent the same vector in the figure. We can also use the notation \(\underline{AB}\). In general in this quiz we will use underlining but we will also use the arrow notation where it is particularly helpful. Note that in these quizzes all math is in boldface so this won't distinguish vectors here but you should be aware of the convention for when you encounter vectors elsewhere.<br></p>
<hr>
<h4 class="HELM_example">Example <br></h4>
<p>The figure below shows an object pulled by a force of \(10\ \mathrm{N}\) at an angle of \(60^{\circ}\) to the horizontal.</p>
<figure class="HELM_figure">
<img src="@@PLUGINFILE@@/9-1-1Ex1ForceOnBlockQ.svg" alt="A an arrow sticks out from a block at an angle of 60 degrees to show the force acting on the block" class="img-responsive atto_image_button_text-bottom" width="195" height="137">
</figure>
<p>Show how this force can be represented by a vector.</p>
<h4 class="HELM_solution">Solution</h4>
<p>The force can be represented by drawing a line of length 10 units at an angle of \(60^\circ\) to the horizontal, as shown below.</p>
<figure class="HELM_figure">
<img src="@@PLUGINFILE@@/9-1-1Ex1Soln.svg" alt="Arrow drawn at 60 degrees to the horizontal with a scale showing that the arrow is 10 units long." class="img-responsive atto_image_button_text-bottom" width="255" height="87">
</figure>
<p>We have labelled the force \(\underline{F}\). When several forces are involved they can be labelled \(\underline{F}_1\), \(\underline{F}_2\) and so on.<br></p>
<hr>
<p>When we wish to refer simply to the magnitude (or length) of a vector we write this using the <span><strong>modulus</strong></span> sign as \(|\overrightarrow{AB}|\), or \(|\underline{a}|\), or simply \(a\) (without the underline.)</p>
<p>In general two vectors are said to be <b>equal vectors</b> if they have the same magnitude and same direction. So, in the figure below the vectors \(\overrightarrow{CD}\) and \(\overrightarrow{AB}\) are equal even though their locations differ.</p>
<figure class="HELM_figure">
<img src="@@PLUGINFILE@@/9-1-1EqualVectors.svg" alt="A pair of vectors that are parallel and of equal length." class="img-responsive atto_image_button_text-bottom" width="319" height="140">
<figcaption>Equal vectors</figcaption>
</figure>
<p>This is a useful and important property of vectors: a vector is defined only by its direction and magnitude, not by its location in space. These vectors are often called <b>free</b> vectors.</p>
<p>The vector \(-\underline{a}\) is a vector in the opposite direction to \(\underline{a}\), but has the same magnitude as \(\underline{a}\), as shown below.</p>
<figure class="HELM_figure">
<img src="@@PLUGINFILE@@/9-1-1OppositeVectors.svg" alt="Two vectors which are parallel and equal in length but the arrows point in opposite directions." class="img-responsive atto_image_button_text-bottom" width="329" height="147">
</figure>
<p>Geometrically, if \(\underline{a}=\overrightarrow{AB}\) then \(-\underline{a}=\overrightarrow{BA}\).</p>]]></text>
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