The components of the vector 𝐮 are one, two as a terminal point of the vector is one unit right of the initial point and two units up from the initial point. What are the components of the vector 𝐯?
The preamble to this question talks about the vector 𝐮, which we can see represented in our diagram. And we’re told that the components of this vector are one, two. The components of a vector are written much like the coordinates of a point. They are two numbers which are separated by a comma.
The difference is that the components of a vector are written inside angled brackets and not parentheses as we have for coordinates. Let’s talk about the numbers inside the brackets, the first number is called the 𝑥-component. Again, the analogy with points and coordinates is clear.
And the 𝑥-component of the vector 𝐮 is one as the terminal point of the vector is one unit right of the Initial point. And by looking at the diagram, we can see that this is true. The second number inside the brackets is called the 𝑦-component of the vector. Its value is two because the terminal point is two units up from the initial point.
And again this is something that we can see by looking at the diagram. We want to find the components of the vector 𝐯, which is represented on our diagram alongside 𝐮. If we let 𝑥 be the 𝑥-component of 𝐯 and 𝑦 be the 𝑦-component of 𝐯, then we can write these components like so. And you can think of these two numbers as giving instructions for how to get from the initial point to the terminal point of the vector.
You go 𝑥 units right and then 𝑦 units up. Let’s write this idea down. This is a similar idea to thinking of the coordinates of a point as being instructions which tell you how to get to the point from the origin. So what are the values of 𝑥 and 𝑦?
Well we can basically read them off from the diagram by counting squares. 𝑥 is two and 𝑦 is five. To get from the initial point of 𝐯 to the terminal point of 𝐯, we need to move two units right and then five units up. Our answer is therefore that the components of the vector 𝐯 are two, five.