<h2>The Resultant Vector</h2>
<p>In previous lessons, you learnt about adding vectors together in one dimension. The same principle can be applied for vectors in two dimensions. The following examples show addition of vectors. Vectors that are parallel can be shifted to fall on a line. Vectors falling on the same line are called <em>co-linear</em> vectors. To add co-linear vectors we use the tail-to-head method you learnt earlier. In the figure below we remind you of the approach of adding co-linear vectors to get a resultant vector.</p>
<div class="wp-caption alignnone" style="width: 536px"><img alt="c28cb99f9ee075db7212dbaa68cad8c6.png" height="359" src="https://data.ulearngo.com/assets/7172fae0-81f4-4904-af59-9b963277ec30" width="536"/><p class="wp-caption-text">Adding co-linear vectors to get a resultant vector.</p></div>
<p>In the above figure the blue vectors are in the \(y\)-direction and the red vectors are in the \(x\)-direction. The two black vectors represent the resultants of the co-linear vectors graphically.</p>
<p>What we have done is implement the tail-to-head method of vector addition for the vertical set of vectors and the horizontal set of vectors.</p>
<div class="ns-school-emp">
<h2>Example 1: Head to Tail Addition in One-Dimension</h2>
<div class="ns-school-defn">
<h3>Question</h3>
<div>
<p>Use the graphical head-to-tail method to determine the resultant force on a rugby player if two players on his team are pushing him forwards with forces of \(\stackrel{\to }{{F}_{1}}\) = \(\text{600}\) \(\text{N}\) and \(\stackrel{\to }{{F}_{2}}\) = \(\text{900}\) \(\text{N}\) respectively and two players from the opposing team are pushing him backwards with forces of \(\stackrel{\to }{{F}_{3}}\) = \(\text{1 000}\) \(\text{N}\) and \(\stackrel{\to }{{F}_{4}}\) = \(\text{650}\) \(\text{N}\) respectively.</p>
</div>
<div>
<h3>Step 1: Choose a scale and a reference direction</h3>
<p>Let's choose a scale of \(\text{100}\) \(\text{N}\): \(\text{0.5}\) \(\text{cm}\) and for our diagram we will define the positive direction as <em>to the right.</em></p>
</div>
<div>
<h3>Step 2: Choose one of the vectors and draw it as an arrow of the correct length in the correct direction</h3>
<p>We will start with drawing the vector \(\stackrel{\to }{{F}_{1}}\) = \(\text{600}\) \(\text{N}\) pointing in the positive direction.Using our scale of \(\text{0.5}\) \(\text{cm}\) : \(\text{100}\) \(\text{N}\), the length of the arrow must be \(\text{3}\) \(\text{cm}\) pointing to the right.</p>
<img alt="1cb205491fc742e7c0f3ebae0b9de03f.png" src="https://data.ulearngo.com/assets/23ba3361-61be-4c2a-af3d-a386e2e67500"/></div>
<div>
<h3>Step 3: Take the next vector and draw it starting at the arrowhead of the previous vector</h3>
<p>The next vector is \(\stackrel{\to }{{F}_{2}}\) = \(\text{900}\) \(\text{N}\) in the same direction as \(\stackrel{\to }{{F}_{1}}\). Using the scale, the arrow should be \(\text{4.5}\) \(\text{cm}\) long and pointing to the right.</p>
<img alt="3695051eaf017ffb0a440598f5a5dd54.png" src="https://data.ulearngo.com/assets/6d3165f8-3666-470b-8206-f53758732ae8"/></div>
<div>
<h3>Step 4: Take the next vector and draw it starting at the arrowhead of the previous vector</h3>
<p>The next vector is \(\stackrel{\to }{{F}_{3}}\) = \(\text{1 000}\) \(\text{N}\) in the <em>opposite</em> direction. Using the scale, this arrow should be \(\text{5}\) \(\text{cm}\) long and point to the <em>left</em>.</p>
<p><strong>Note:</strong> We are working in one dimension so this arrow would be drawn on top of the first vectors to the left. This will get confusing so we'll draw it next to the actual line as well to show you what it looks like.</p>
<img alt="a20be69c2ecf780b955d9142eea6fd7d.png" src="https://data.ulearngo.com/assets/f5b7b2af-07f2-4e47-aec5-1b6ca5583068"/></div>
<div>
<h3>Step 5: Take the next vector and draw it starting at the arrowhead of the previous vector</h3>
<p>The fourth vector is \(\stackrel{\to }{{F}_{4}}\) = \(\text{650}\) \(\text{N}\) in the opposite direction. Using the scale, this arrow must be \(\text{3.25}\) \(\text{cm}\) long and point to the left.</p>
<img alt="7b007b482a7f8ea26aa958a9466ca109.png" src="https://data.ulearngo.com/assets/3036b5bf-19a3-44e7-94d7-2323c40151dc"/></div>
<div>
<h3>Step 6: Draw the resultant, measure its length and find its direction</h3>
<p>We have now drawn all the force vectors that are being applied to the player. The resultant vector is the arrow which starts at the tail of the first vector and ends at the head of the last drawn vector.</p>
<img alt="2a3105494323df985eaf5f09ba10fe16.png" src="https://data.ulearngo.com/assets/ac68c9e3-4481-4c5e-afa2-90c6298eb226"/>
<p>The resultant vector measures \(\text{0.75}\) \(\text{cm}\) which, using our scale is equivalent to \(\text{150}\) \(\text{N}\) and points to the left (<em>or</em> the negative direction <em>or</em> the direction the opposing team members are pushing in).</p>
</div>
</div>
</div>
<section></section>

1c4ddb66-ad36-4f81-90e7-4629f497d0b4

Resultant Vector Continued

Physics is a science discipline concerned with the nature and properties of matter and energy. Physics topics include mechanics, heat, light and radiation, sound, electricity, magnetism, the structure of atoms, and so on.

Physics

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.


Resultant Vector Continued

Track Your Learning Progress