So we know two things about vectors. First of all, a vector has length and direction and that means you can translate it to any other point in the plane and it’s still the same vector. And second, we learnt that the components of a vector drawn from the origin to point p, are exactly the coordinates of point p.
So let’s use that here to find the component form of any vector QR. So I have QR drawn here from one point to another any two points. Let me observe that this horizontal length here would be x2 minus x1. And this vertical length would be y2 minus y1.
Now we translate this vector down to the origin so that its initial point is at the origin. These lengths are still going to be the same this is still going to minus x2 minus x1, y2 minus y1. And that means that the coordinates of this point would be x2 minus x1 and y2 minus y1. And that means these are the components of this vector, but they are the same vector. So the components are going to be x2 minus x1, y2 minus y1. So it’s really easy to come up with the components of a vector from one point to another.
Just literally subtract like components. Make sure you always subtract in the same order though. Let’s take a look at an example. It says determine the components of PQ and express it in the form <a,b> with these corner brackets.
All right here’s P and q, so PQ, vector PQ, remember it starts at point Q and ends at point q. So you have to subtract from Q to p. So it will be 6 minus 2, 11 minus 9. And that’s <4,2>. PQ here, 4 minus 4, 8 minus -1, 8 plus 1. It’s going to be <0,9>. PQ here -8 minus -5, -8 plus 5 is -3. -7 minus -17 is -7, plus 17, 10. And here, 0 minus 6, -6, 5 minus 0, 5. Always remember you got to subtract the same order, so second minus first second minus first.
That’s how you find the components of any vector from one point to another.