<h2>Magnitude of the Resultant of Vectors at Right Angles</h2>
<p>We apply the same principle to vectors that are at right angles or perpendicular to each other.</p>
<section>
<h2 class="title">Sketching Tail-to-Head Method</h2>
<p>The tail of the one vector is placed at the head of the other but in two dimensions the vectors may not be co-linear. The approach is to draw all the vectors, one at a time. For the first vector begin at the origin of the Cartesian plane, for the second vector draw it from the head of the first vector. The third vector should be drawn from the head of the second and so on. Each vector is drawn from the head of the vector that preceded it. The order doesn't matter as the resultant will be the same if the order is different.</p>
<p>Let us apply this procedure to two vectors:</p>
<ul>
<li>\(\vec{F}_{1} = \text{2}\text{ N}\) in the positive \(y\)-direction</li>
<li>\(\vec{F}_{2} = \text{1.5}\text{ N}\) in the positive \(x\)-direction</li>
</ul>
<p>We first draw a Cartesian plane with the first vector originating at the origin:</p>
<img alt="a32e46ad4db9d0c1cdeb5a6d0314d2e5.png" src="https://data.ulearngo.com/assets/e611a5f1-487b-4af7-ac1a-12543ae87f3b"/>
<p>The next step is to take the second vector and draw it from the head of the first vector:</p>
<img alt="e1facdc27038a359c3c65b7b15258ade.png" src="https://data.ulearngo.com/assets/db4dd259-fee7-44e5-9952-8124d2dbe102"/>
<p>The resultant, \(\vec{R}\), is the vector connecting the tail of the first vector drawn to the head of the last vector drawn:</p>
<img alt="1e7f382fe09e3d82c1d0643ed7921ff1.png" src="https://data.ulearngo.com/assets/fe82a1ef-96cd-47cd-a30b-b7de861de6c6"/>
<p>It is important to remember that the order in which we draw the vectors doesn't matter. If we had drawn them in the opposite order we would have the same resultant, \(\vec{R}\). We can repeat the process to demonstrate this:</p>
<p>We first draw a Cartesian plane with the second vector originating at the origin:</p>
<img alt="947d14e1109736257a81badc8e113207.png" src="https://data.ulearngo.com/assets/049e4061-8663-4f84-bb74-b33028631c90"/>
<p>The next step is to take the other vector and draw it from the head of the vector we have already drawn:</p>
<img alt="2dc27c7471b88957e5b85e5795013932.png" src="https://data.ulearngo.com/assets/f60c8e81-63f5-44b2-92de-d1682d71d2ec"/>
<p>The resultant, \(\vec{R}\), is the vector connecting the tail of the first vector drawn to the head of the last vector drawn (the vector from the start point to the end point):</p>
<img alt="2452c7156241f9222b67fe8a66d4b4c2.png" src="https://data.ulearngo.com/assets/e562f039-914f-47dd-a47a-c44da6ff3a9f"/>
<div class="ns-school-emp">
<h2>Example 1: Sketching Vectors Using Tail-to-Head</h2>
<div class="ns-school-defn">
<h3>Question</h3>
<div>
<p>Sketch the resultant of the following force vectors using the tail-to-head method:</p>
<ul>
<li>\(\vec{F}_{1} = \text{2}\text{ N}\) in the positive \(y\)-direction</li>
<li>\(\vec{F}_{2} = \text{1.5}\text{ N}\) in the positive \(x\)-direction</li>
<li>\(\vec{F}_{3} = \text{1.3}\text{ N}\) in the negative \(y\)-direction</li>
<li>\(\vec{F}_{4} = \text{1}\text{ N}\) in the negative \(x\)-direction</li>
</ul>
</div>
<div>
<h3>Step 1: Draw the Cartesian plane and the first vector</h3>
<p>First draw the Cartesian plane and force, \(\vec{F}_{1}\) starting at the origin:</p>
<img alt="6d845a34bc851dd4f5f1e63519573df1.png" src="https://data.ulearngo.com/assets/c86e3257-a850-4adc-95e1-63fc965f8d97"/></div>
<div>
<h3>Step 2: Draw the second vector</h3>
<p>Starting at the head of the first vector we draw the tail of the second vector:</p>
<img alt="998b6fa84fd770d251b0bc648840fa43.png" src="https://data.ulearngo.com/assets/dd6a8e61-7458-4348-929b-0ccc2620d69d"/></div>
<div>
<h3>Step 3: Draw the third vector</h3>
<p>Starting at the head of the second vector we draw the tail of the third vector:</p>
<img alt="56d18f851f1729eba09e9294f725da5e.png" src="https://data.ulearngo.com/assets/c6b88a88-4301-4c1c-a793-d2d853531df4"/></div>
<div>
<h3>Step 4: Draw the fourth vector</h3>
<p>Starting at the head of the third vector we draw the tail of the fourth vector:</p>
<img alt="431d3269b9dc79c41147c408fb7230d2.png" src="https://data.ulearngo.com/assets/46b39697-0154-4cc4-a8c8-a18072e648a3"/></div>
<div>
<h3>Step 5: Draw the resultant vector</h3>
<p>Starting at the origin draw the resultant vector to the head of the fourth vector:</p>
<img alt="275daa2dc03fdfcc9fffe131d7d9dacc.png" src="https://data.ulearngo.com/assets/b4a28034-bb2e-4f78-b253-42762f100755"/></div>
</div>
</div>
<div class="ns-school-emp">
<h2>Example 2: Sketching Vectors Using Tail-to-Head</h2>
<div class="ns-school-defn">
<h3>Question</h3>
<div>
<p>Sketch the resultant of the following force vectors using the tail-to-head method by first determining the resultant in the \(x\)- and \(y\)-directions:</p>
<ul>
<li>\(\vec{F}_{1} = \text{2}\text{ N}\) in the positive \(y\)-direction</li>
<li>\(\vec{F}_{2} = \text{1.5}\text{ N}\) in the positive \(x\)-direction</li>
<li>\(\vec{F}_{3} = \text{1.3}\text{ N}\) in the negative \(y\)-direction</li>
<li>\(\vec{F}_{4} = \text{1}\text{ N}\) in the negative \(x\)-direction</li>
</ul>
</div>
<div>
<h3>Step 1: First determine \(\vec{R}_{x}\)</h3>
<p>First draw the Cartesian plane with the vectors in the \(x\)-direction:</p>
<img alt="5e44d123b17423ebd8dc6b52f1a96aa1.png" src="https://data.ulearngo.com/assets/ab4e6cac-2549-4fe1-9243-f4838c7b9af4"/></div>
<div>
<h3>Step 2: Secondly determine \(\vec{R}_{y}\)</h3>
<p>Next we draw the Cartesian plane with the vectors in the \(y\)-direction:</p>
<img alt="2d16c08364832592d792a8c1a22b32c7.png" src="https://data.ulearngo.com/assets/18cb52fe-398c-404e-a937-f5e657e4bacd"/></div>
<div>
<h3>Step 3: Draw the resultant vectors, \(\vec{R}_{y}\) and \(\vec{R}_{x}\) head-to-tail</h3>
<img alt="95ecea8ebb22bcafcc911a0599bc204b.png" src="https://data.ulearngo.com/assets/80718949-c636-4e0c-b2b6-e9c15c6a63c2"/></div>
<div>
<h3>Step 4: Comparison of results</h3>
<p>To double check, we can replot all the vectors again as we did in the previous worked example to see that the outcome is the same:</p>
<img alt="86dde2e4c0ad507c69a92426abbf6778.png" src="https://data.ulearngo.com/assets/0f8ba662-a158-4ffa-91de-a119faeb453b"/></div>
</div>
</div>
</section>
<section></section>

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Magnitude of the Resultant

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Magnitude of the Resultant

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