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Magnitude of a vector from initial & terminal points

Video transcript
So, let's say we have vector w, and we're given some information about vector w. We're told what its initial point is, it's the point negative seven comma three, and we know what its terminal point is, it's two comma negative one. And from this, I want to figure out the magnitude of vector w. So, I want to figure out the magnitude of vector w. What is that going to be equal to? And like always, pause the video and see if you can work through it. So I'm going to tackle it, first just by using these numbers, and then I'll draw it out to show why it visually makes sense. So, one way to think about this, is the magnitude... Let me do this in another color just for fun. So... it's the same color. So the magnitude of a vector is going to be its change in x squared plus its change in y squared. So what's it's change in x going to be? So change... do that in green. So our change in x. Well, our change in x is going to be our terminal x point minus our initial x point. So our change in x is going to be two minus negative seven. Two minus negative seven, which is the same thing as two plus seven, which is equal to nine. And that makes sense. To go from negative seven to two, you have to increase by seven just to go to zero and then you have to increase by two more to get to positive two. So you have to increase by nine. So our change in x is a positive nine. Well what's our change in y going to be, our change in y? Well it's going to be our final y, negative one, minus our initial y, minus three, so that is negative four. Now let's make sure that makes sense. If I start at three, I'm going to have to go down to get to negative one, so the negative makes sense, I'm going down. And how much do I have to go down? I have to go down three just to get to zero, and then I have to go down one more to get to negative one. So I have to go down four. So our change in y is negative four. And so, what is the magnitude going to be? Well, it's going to be equal to the square root of delta x squared... actually let me just write it out, I don't want to skip any steps. So nine squared plus negative four squared, Which is equal to... This is equal to 36... let me do it in green color... 36 plus 16, which is equal to... that's going to be equal to the square root of 52. And to see why that makes sense, and once again I was able to do this without having to draw anything, but to see why that makes sense, there's a couple of ways you could think about it. Now that we know the change in x and change in y, we can actually write the vector as its components, so we could say that vector w is equal to the vector. Its change in x is positive nine, so its x component is nine, and its change in y is negative four, so its y component is negative four, and from that you could figure out its magnitude by saying "OK, the square root of the x component squared plus the y component squared, well, that's going to be the square root of 52." And if you want to see it visually... Let me make a plot here. So that is my y-axis, and I have the point negative seven comma three, which is gonna be up here, and I go to two negative one, so let me draw the x-axis here. So that's the x-axis. Negative seven comma three, the initial point. Negative one, negative two, negative three, negative four, negative five, negative six, negative seven, and then one, two, three. So our initial point is going to be right about there. That's the point negative seven comma three. And then our terminal point... our terminal point I'll do in mauve. Our terminal point, we go one, two and then we go negative one. So it's gonna be right over there. That's the point two comma negative one. And then we can draw our vector going from our initial point to our terminal point, just like that. And so what we just figured out, the square root of 52, that's the length of this arrow, square root of 52. And once again, it comes straight out of the distance formula between these two points, which comes straight out of the Pythagorean Theorem. Our change in x, which was that nine, that is going from an x value of negative seven to an x value of positive two, that was where the nine comes from. And that's also our x component. And then our change in y, which you could also view as our y component, we go from y is equal to three to y... actually let me draw it it a little bit... y is equal to three to y is equal to negative one. So that's where that negative four came from. And so, if you just wanted to think about the absolute distances of these sides, it's nine and four, Pythagorean Theorem, this is a right triangle, nine squared plus four squared is going to be the hypotenuse squared. So the hypotenuse squared is going to be 52, or the length of the hypotenuse is the square root of 52.