<h2>Algebraic Techniques: Vectors in a Straight Line</h2>
<section>
<p>Whenever you are faced with adding vectors acting in a straight line (i.e. some directed left and some right, or some acting up and others down) you can use a very simple algebraic technique:</p>
<h2><strong>Method: Addition/Subtraction of Vectors in a Straight Line</strong></h2>
<ol>
<li>Choose a positive direction. As an example, for situations involving displacements in the directions west and east, you might choose west as your positive direction. In that case, displacements east are negative.</li>
<li>Next simply add (or subtract) the magnitude of the vectors using the appropriate signs.</li>
<li>As a final step the direction of the resultant should be included in words (positive answers are in the positive direction, while negative resultants are in the negative direction).</li>
</ol>
<p>Let us consider a few examples.</p>
<div class="ns-school-emp">
<h2>Example 1:</h2>
<div class="ns-school-defn">
<h3>Question</h3>
<div>
<p>A tennis ball is rolled towards a wall which is $\text{10}$ $\text{m}$ away from the ball. If after striking the wall the ball rolls a further $\text{2.5}$ $\text{m}$ along the ground away from the wall, calculate algebraically the ball's resultant displacement.</p>
</div>
<div>
<h3>Step 1: Draw a rough sketch of the situation</h3>
<img alt="50bcb1da2c854148beee0d4e3b4ecb17.png" src="https://data.ulearngo.com/assets/3597bdbc-466f-4803-b405-cb3f490a0f13"/></div>
<div>
<h3>Step 2: Decide which method to use to calculate the resultant</h3>
<p>We know that the resultant displacement of the ball ($\vec{x_{R}}$) is equal to the sum of the ball's separate displacements ($\vec{x_{1}}$ and $\vec{x_{2}}$):</p>

$\vec{x_{R}} = \vec{x_{1}} + \vec{x_{2}}$

<p>Since the motion of the ball is in a straight line (i.e. the ball moves towards and away from the wall), we can use the method of algebraic addition just explained.</p>
</div>
<div>
<h3>Step 3: Choose a positive direction</h3>
<p>Let's choose the <strong>positive</strong> direction to be towards the wall. This means that the <strong>negative</strong> direction is away from the wall.</p>
</div>
<div>
<h3>Step 4: Now define our vectors algebraically</h3>
<p>With right positive:</p>

\begin{align*} \vec{x_{1}} &amp; = + \text{10,0}\text{ m} \\ \vec{x_{2}} &amp; = -\text{2,5}\text{ m} \end{align*}</div>
<div>
<h3>Step 5: Add the vectors</h3>
<p>Next we simply add the two displacements to give the resultant:</p>

\begin{align*} \vec{x_{R}} &amp; = \left(+\text{10,0}\text{ m}\right) + \left(-\text{2,5}\text{ m}\right) \\ &amp; = \left(+\text{7,5}\text{ m}\right) \end{align*}</div>
<div>
<h3>Step 6: Quote the resultant</h3>
<p>Finally, in this case <u>towards the wall is the positive direction</u>, so: $\vec{x_{R}} = \text{7.5}\text{ m}$ towards the wall.</p>
</div>
</div>
</div>
<div class="ns-school-emp">
<h2>Example 2:</h2>
<div class="ns-school-defn">
<h3>Question</h3>
<div>
<p>Suppose that a tennis ball is thrown horizontally towards a wall at an initial velocity of $\text{3}$ $\text{m·s$^{-1}$}$ to the right. After striking the wall, the ball returns to the thrower at $\text{2}$ $\text{m·s$^{-1}$}$. Determine the change in velocity of the ball.</p>
</div>
<div>
<h3>Step 1: Draw a sketch</h3>
<p>A quick sketch will help us understand the problem.</p>
<img alt="1fc3f7dd9dc824176ec3671c6733a76a.png" src="https://data.ulearngo.com/assets/39c65879-2af1-472e-bc38-aaec5a45f48a"/></div>
<div>
<h3>Step 2: Decide which method to use to calculate the resultant</h3>
<p>Remember that velocity is a vector. The change in the velocity of the ball is equal to the difference between the ball's initial and final velocities:</p>

$\Delta \vec{v} = \vec{v_{f}} - \vec{v_{i}}$

<p>Since the ball moves along a straight line (i.e. left and right), we can use the algebraic technique of vector subtraction just discussed.</p>
</div>
<div>
<h3>Step 3: Choose a positive direction</h3>
<p>Choose the <strong>positive</strong> direction to be towards the wall. This means that the <strong>negative</strong> direction is away from the wall.</p>
</div>
<div>
<h3>Step 4: Now define our vectors algebraically</h3>

\begin{align*} \vec{v_{i}} &amp; = +\text{3}\text{ m·s$^{-1}$} \\ \vec{v_{f}} &amp; = -\text{2}\text{ m·s$^{-1}$} \end{align*}</div>
<div>
<h3>Step 5: Subtract the vectors</h3>
<p>Thus, the change in velocity of the ball is:</p>

\begin{align*} \Delta \vec{v} &amp; = \left(-\text{2}\text{ m·s$^{-1}$}\right) - \left(+\text{3}\text{ m·s$^{-1}$}\right) \\ &amp; = -\text{5}\text{ m·s$^{-1}$} \end{align*}</div>
<div>
<h3>Step 6: Quote the resultant</h3>
<p>Remember that in this case <u>towards the wall means a positive velocity</u>,</p>
<p>so <u>away from the wall means a negative velocity</u>: $\Delta \vec{v}=\text{5}\text{ m·s$^{-1}$}$ away from the wall.</p>
</div>
</div>
</div>
<div class="ns-school-emp">
<h2>Example 3:</h2>
<div class="ns-school-defn">
<h3>Question</h3>
<div>
<p>A man applies a force of $\text{5}$ $\text{N}$ on a crate. The crate pushes back on the man with a force of $\text{2}$ $\text{N}$. Calculate algebraically the resultant force that the man applies to the crate.</p>
</div>
<div>
<h3>Step 1: Draw a sketch</h3>
<p>A quick sketch will help us understand the problem.</p>
<img alt="92d42315f123ddc745e1fb4f70032bfc.png" src="https://data.ulearngo.com/assets/cc61ade7-a03e-4359-a077-02fa6c065e40"/></div>
<div>
<h3>Step 2: Decide which method to use to calculate the resultant</h3>
<p>Remember that force is a vector. Since the crate moves along a straight line (i.e. left and right), we can use the algebraic technique of vector addition just discussed.</p>
</div>
<div>
<h3>Step 3: Choose a positive direction</h3>
<p>Choose the <strong>positive</strong> direction to be towards the crate (i.e. in the same direction that the man is pushing). This means that the <strong>negative</strong> direction is away from the crate (i.e. against the direction that the man is pushing).</p>
</div>
<div>
<h3>Step 4: Now define our vectors algebraically</h3>

\begin{align*} \vec{F_{\text{man}}} &amp; = +\text{5}\text{ N} \\ \vec{F_{\text{crate}}} &amp; = -\text{2}\text{ N} \end{align*}</div>
<div>
<h3>Step 5: Subtract the vectors</h3>
<p>Thus, the resultant force is:</p>

\begin{align*} \vec{F_{\text{man}}} + \vec{F_{\text{crate}}} &amp; = \left(\text{5}\text{ N}\right) + \left(\text{2}\text{ N}\right) \\ &amp; = \text{7}\text{ N} \end{align*}</div>
<div>
<h3>Step 6: Quote the resultant</h3>
<p>Remember that in this case <u>towards the crate means a positive force</u>: $\text{7}$ $\text{N}$ towards the crate.</p>
</div>
</div>
</div>
<p>Remember that the technique of addition and subtraction just discussed can only be applied to vectors acting along a straight line. When vectors are not in a straight line, i.e. at an angle to each other then simple geometric and trigonometric techniques can be used to find resultant vectors.</p>
</section>

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Algebraic Techniques of Vector Addition

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Algebraic Techniques of Vector Addition

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