Example of vector magnitude from initial and terminal points

What we have depicted here we could call vector w, and you can see from this diagram that its initial point is right over here. It's the point negative seven, comma, positive three, and its terminal point is this point right over here, which is the point two, comma, negative one.

What I want to do in this video is think about what is the magnitude of our vector. If you're saying, "What do I mean by magnitude?" well, one way to think about it is, what is the length of this vector? How long is it? Pause this video and see if you can figure it out based on the information that's given.

Well, one thing that might jump out at you is that the magnitude of this vector, the length of this vector, is really just the distance between these two points. So if you want the magnitude, you just have to apply essentially the distance formula here, which is essentially just the Pythagorean theorem.

What we could do is construct a right triangle. I will do that like this. So this height in red, that would be our change in y; that would be our change in y. And then what I am doing in this light blue color, this would be our change in x, change in x.

We know from the Pythagorean theorem that the length of the hypotenuse, which would be the magnitude of our vector, that that is going to be equal to the square root of our change in x squared plus change in y squared. And so what will this be?

Well, what is our change in x? Our change in x you could view it as your x final minus x initial. So this would be 2 minus negative 7. So this is 2 minus negative 7, which is equal to positive 9. And so this would be 9 squared.

Then what is our change in y? Our change in y you could view this as your y final, which is negative 1, minus your y initial, which is 3. Three minus three, which is equal to negative four, and you did indeed go down by four. So this is going to be negative four.

And so our magnitude is going to be equal to the square root of nine squared, which is 81, plus negative 4 squared, which is 16. And so what is that going to be? Let's see if you add 6 to it, that gets to 87, and you have another 10, the square root of 97.

So this is going to be equal to the square root of 97, which I don't think can be simplified anymore. But if you wanted to estimate what that is, that's almost the square root of 100. So this number is going to be a little bit less than 10 is the magnitude of this vector.

In this case, we were able to do that from its initial point and its ending point. Now, another way that a vector might be specified, they might just be given an x component and a y component.

So, for example, in this situation, you could actually define our vector w by the sum of two vectors. One of which is, let me just send the blue color, one of which is the x component. So you could view this as the x component of w.

Then the other is the y component. You could view this as the y component of w. And you can immediately see that that y component is the same as our change in y, and the x component is the same thing as our change in x.

So sometimes you will see something like this: the vector w is equal to, and it might look like coordinates, but they're really giving you the components. So the x component is positive 9, the x component is positive 9, and then the y component is negative 4; it is negative 4.

Now, you might say, "Hey, well with something like this, all I know is the x and y component. I don't know where it exactly starts or ends." And that's actually on purpose because a vector, you only care about the magnitude and the direction, and this is actually specifying both.

If you wanted the magnitude here, you just take the square root of the sum of the squares of the magnitudes. So once again, the square root of 9 squared plus negative 4 squared is going to be the square root of 97.

So you want the magnitude and the direction, which this will specify, but you can shift it around all that you want. This vector w, you could also have it starting right over here and going 9 in the positive x direction and then negative 4 in the positive y direction, or negative 4 down.

So it might look something like this. And so, once again, you can shift vectors around. You care about magnitude and direction, but hopefully this gives you a sense of how to find magnitude given the components or given the starting and ending points.