Video: The Components of a Vector on a Grid

Video Transcript

The components of the vector 𝐮 are negative one, negative two. As the terminal point of the vector is negative one units right or one unit left and negative two units up or two units down from the initial point. What are the components of the vector 𝐯?

So this question begins by explaining to us how to find the components of the vector 𝐮 in the diagram. The terminal point is where the vector ends, and the initial point is where the vector begins. And we can use the arrow on the vector to determine its direction.

The arrow on the vector 𝐮 is pointing downwards to the left. So the initial point is the point labeled in orange, and the terminal point is the point labeled in pink. We see that, to move from the initial point to the terminal point, we have to move one unit to the left and then two units downwards. We can also think of this as moving negative one units right and negative two units up. This is because the convention when writing the components of a vector is to list the number of units right first and then the number of units up. So movements to the left and movements down are expressed using negative values.

Let’s now consider this vector 𝐯. We can see that the arrow on this vector is again pointing downwards to the left. So the initial point is the point on the top right, and the terminal point is the point on the bottom left. To move from the initial point to the terminal point, we move one, two units left first of all. And we then have to move one, two, three, four units down.

But remember, when we describe this as a vector, we need to describe the motion right and up. So two units left can be written as negative two units right, and four units down can be written as negative four units up. Using this convention then, the components of the vector 𝐯 will be negative two, negative four.

There is actually a special relationship between the vectors 𝐮 and 𝐯 in this question. If I draw the vector 𝐮 right next to the vector 𝐯, we can see that these two vectors are parallel. They’re traveling in the same direction. But the vector 𝐯 is twice the vector 𝐮. Two lots of the vector 𝐮 make up the vector 𝐯. So we could also answer this question by saying that the vector 𝐯 is equal to twice the vector 𝐮. That’s two multiplied by the vector negative one, negative two.

To multiply a vector by a scalar — that’s just a number, in this case two — we multiply each of the components by that scalar. So the first component will be two multiplied by negative one, and the second component will be two multiplied by negative two. That gives the vector with components negative two, negative four, which is what we’ve already found the vector 𝐯 to be using the diagram. So by considering two approaches, we have our answer to the problem. The vector 𝐯 in component form is negative two, negative four.