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Vector magnitude from initial & terminal points

Video transcript
- [Instructor] 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? And 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. And so if you want the magnitude, you just have to apply essentially the distance formula here, which is essentially just the Pythagorean theorem. So 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. And 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, that's going to be equal to the square root of our change in x squared, change in x squared, plus change in y squared, plus change in y squared. And so, what will this be? Well, what is our change in x? Our change in x, our change in x, you could view it as your x final minus x initial. So this would be two minus negative seven. So this is two minus negative seven, which is equal to positive nine. And so this would be nine squared. And then what is our change in y? Our change in y, you could view this as your y final, which is negative one, minus your y initial, which is 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 is 81, plus negative four squared is 16. And so what is that going to be? Let's see, if you add six to it, that's gets to 87, and you add 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. And in this case, we were able to do that from its initial points 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. And 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 do this in the blue color, one of which is the x-component. So you could view this as the x-component of w, and then the other is the y-component. You could view this as, you could view this as the y-component of w. And you could 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. And so sometimes you will see something like this. The vector, 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 nine. The x-component is positive nine, and then the y-component is negative four. It is negative four. 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 and 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'd just take the square root of the sum of the squares of the magnitudes. So once again, the square root of nine squared plus negative four 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, you could also have it starting right over here and going nine in the positive x-direction and then negative four in the positive y-direction and, or negative four down. And 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 sense of how to find magnitude given the components or given the starting and ending points.