<h2>Vector Addition Using Components</h2>
<p>Components can also be used to find the resultant of vectors. This technique can be applied to both graphical and algebraic methods of finding the resultant. The method is straightforward:</p>
<ol>
<li>make a rough sketch of the problem;</li>
<li>find the horizontal and vertical components of each vector;</li>
<li>find the sum of all horizontal components, \(\vec{R}_x\);</li>
<li>find the sum of all the vertical components, \(\vec{R}_y\);</li>
<li>then use them to find the resultant, \(\vec{R}\).</li>
</ol>
<p>Consider the two vectors, \(\vec{F}_1\) and \(\vec{F}_2\), in the figure below, together with their resultant, \(\stackrel{\to }{R}\).</p>
<div class="wp-caption alignnone" style="width: 536px"><img alt="7a6d47d7e03cdb49ba29cd7455e02386.png" height="418" src="https://data.ulearngo.com/assets/9179b6f2-0abb-49c9-a3b3-562714cf4dfd" width="536"/><p class="wp-caption-text">An example of two vectors being added to give a resultant.</p></div>
<p>Each vector in the figure above can be broken down into one component in the \(x\)-direction (horizontal) and one in the \(y\)-direction (vertical). These components are two vectors which when added give you the original vector as the resultant. This is shown in the figure below:</p>
<div class="wp-caption alignnone" style="width: 630px"><img alt="6578e1f4572539b72a487b82b9805200.png" height="465" src="https://data.ulearngo.com/assets/30da693b-83a8-4b98-8895-d36f52092e3d" width="630"/><p class="wp-caption-text">Adding vectors using components.</p></div>
<p>We can see that: \begin{align*} \vec{F}_1 &amp;= \vec{F}_{1x} + \vec{F}_{1y} \\ \vec{F}_2 &amp;= \vec{F}_{2x} + \vec{F}_{2y} \\ \vec{R} &amp;= \vec{R}_x + \vec{R}_y \end{align*} \begin{align*} \text{But,}\ \vec{R}_x &amp;= \vec{F}_{1x} + \vec{F}_{2x} \\ \text{and}\ \vec{R}_y &amp;= \vec{F}_{1y} + \vec{F}_{2y} \end{align*}</p>
<p>In summary, addition of the \(x\)-components of the two original vectors gives the \(x\)-component of the resultant. The same applies to the \(y\)-components. So if we just added all the components together we would get the <strong>same answer!</strong> This is another important property of vectors.</p>
<h2 class="note visit">Optional Video on Vectors </h2>
<div class="note visit"><iframe allowfullscreen="allowfullscreen" frameborder="0" height="360" src="https://www.youtube.com/embed/pXYlOC6u9w8?rel=0&amp;modestbranding=1&amp;cc_lang_pref=en&amp;cc_load_policy=1" width="640"></iframe></div>

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Vector Addition Using Components

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Discover the principles and operations of scalars and vectors such as graphical and algebraic techniques of vector addition, finding magnitude with Pythagoras theorem, and resultant force on a submarine.


Vector Addition Using Components

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