Question Video: Finding a Scalar Multiple of a Vector Graphically Mathematics • Third Year of Secondary School

Video Transcript

Vector 𝐀 is represented by the following graph. Which of the following represents a half 𝐀?

In this question, we’re given a graphical representation of a vector 𝐀. We need to determine which of five given options is the correct graphical representation of a half 𝐀. And there are a few different ways we could answer this question. We’re going to go through two of these.

Let’s start by recalling what it means to multiply a vector by a scalar. And there’s two different ways of thinking about multiplying a vector by a scalar. Let’s start with the graphical interpretation. If we have a vector 𝐯 and we multiply this by a scalar 𝑘, then the length of our vector will be multiplied by 𝑘. If 𝑘 is positive, the direction of this vector stays exactly the same. However, if 𝑘 is negative, then we flip the direction of the vector. And it’s worth noting if 𝑘 is equal to zero, we’re multiplying a vector by zero. This gives us the zero vector.

And we can use this to answer our question. We’re given a graphical representation of 𝐀, and we want to determine a half times 𝐀. So our value of 𝑘 is one-half, which is positive. This means we need to find the vector of one-half length of 𝐀, which points in the same direction. And we can do this directly from the diagram. There’s a few different ways we could do this. One way is to note that the midpoint of the line segment from zero, zero to two, two is the point one, one. This means it cuts the vector in half. So, for example, the vector from zero, zero to one, one is the vector one-half 𝐀.

And it’s also worth pointing out here vectors are only defined by their magnitude and direction. So we can draw these vectors anywhere in the plane. For example, we could’ve drawn the vector from one, one to two, two. This would also be the vector one-half 𝐀, since it has the same magnitude and direction as the vector from zero, zero to one, one. Similarly, we could’ve also drawn our vector anywhere else in the plane. As long as the vectors have the same magnitude and direction, the vectors will be equal. And we can see that this is only true for the graph in option (B). And this allows us to conclude the answer is option (B).

However, there is a second method we could’ve used to answer this question. So let’s go through this method now. We’ll start by clearing some space. We can then recall a second method we can use to multiply a vector by a scalar. This is by using the components of the vector. 𝑘 times the vector 𝑣 sub one, 𝑣 sub two is equal to the vector 𝑘 times 𝑣 sub one, 𝑘 times 𝑣 sub two. In other words, to multiply a vector by a scalar, we just multiply all of the components of the vector by the scalar. So another method of determining the correct graph will be to find the components of vector 𝐀 and then multiply each component by one-half. We can then sketch the vector one-half 𝐀. And we can determine the components of vector 𝐀 from the graph.

And there’s two different ways of doing this. One way is to consider the horizontal and vertical displacement from the initial point of the vector to its terminal point. In this case, we move two units to the right and then two units upwards. This tells us that 𝐀 is the vector two, two. However, a second way of doing this is to use the fact that we know the coordinates of the initial point and terminal points of vector 𝐀. We can just recall that each component of a vector is equal to the difference in each coordinate of its terminal point and its initial point. So 𝐀 is the vector two minus zero, two minus zero, which simplifies to give us the vector two, two.

We can now use this to determine one-half times vector 𝐀. We need to multiply each component of vector 𝐀 by one-half. This gives us the vector one-half times two, one-half times two. And one-half times two is one. So one-half 𝐀 is the vector one, one. And now since all five of the given options start at the origin, let’s sketch the vector one-half 𝐀 starting at the origin. Its 𝑥-component is one, and its 𝑦-component is also one. So we travel one unit to the right and one unit up to the point with coordinates one, one. And this also confirms that our answer is option (B). Therefore, if 𝐀 is represented by the given graph, we were able to show that only option (B) represents the vector one-half 𝐀.