Video: The Components of a Vector

Consider the vector in the given diagram. What are the coordinates of its terminal point? What are the coordinates of its initial point? What are the components of the vector?

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Video Transcript

This is a multipart question about a vector on a diagram. Consider the vector in the given diagram. What are the coordinates of its terminal point? What are the coordinates of its initial point? And what are the components of the vector?

The first part asks about the terminal point of the vector; this is the point which the vector is pointing to — the end of the vector. We can see that the 𝑥-coordinate of this point is five and the 𝑦-coordinate is one. And so the coordinates of the terminal point are 5, 1. It’s a very similar story with the initial point. This is the point from which the vector starts or comes from. And we can see that its 𝑥-coordinate is one and its 𝑦-coordinate is two.

Finally, what are the components of the vector? There are two components of the vector, and it’s written very much like a point is, except instead of using parentheses, we have angled brackets. The first number tells us how far right of the initial point the terminal point is. By counting the squares, we can see that this is four.

The second component tells us how far up the vector is pointing — how far up the terminal point is from the initial point. We have to go down by one unit, which is the same as going up by negative one unit. Our second component is therefore negative one. So the components of our vector are four, negative one.

Can you see the link between the components of the vector and the coordinates of the terminal and initial point? The first coordinate of the terminal point five minus the first coordinate of the initial point one is equal to the first component of the vector four. Likewise, the second coordinate of the terminal point, the 𝑦-coordinate one, minus the second coordinate of the initial point, the 𝑦-coordinate two, is equal to the second component of the vector, negative one.

This isn’t a coincidence. To get from the initial point to the terminal point, you have to move from the 𝑥-coordinate one to the 𝑥-coordinate to five, moving five minus one equals four units to the right. And you have to move from a 𝑦-coordinate of two to a 𝑦-coordinate of one, which means moving one minus two equals negative one unit upwards or one unit downwards.

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