Magnitude of vectors
Vector magnitude from graph
- [Voiceover] We've already seen that a vector is defined by both its magnitude and its direction. What I want to do in this video is get some practice calculating or figuring out the magnitudes of vectors and I have a vector right over here, vector u, it is, it is, visually depicted here on our corded plane and I wanna figure out its magnitude and I encourage you to pause the video and see if you can figure it out. Well the magnitude of a vector, just the length of this line you could view it as the distance between the initial point and the terminal point right over there and so one way to think about it, you could view this, the magnitude of the vector, let me write this. The magnitude of the vector. Well you could either think of it as the distance formula, which is really, just comes straight out of the Pythagorean theorem. It's going to be the square root of our change in x squared change in x squared plus change in y squared plus change in y squared. Once again this triangle, Greek letter delta is just shorthand for change in y and this once again comes straight out of the distance formula which really comes out of the Pythagorean theorem. It'll become a little bit more obvious when I draw the change in y and I draw the change in x on this diagram. So what's our change in y? Well we're starting at our initial point. We're starting at y is equal to nine and we are, to get to the y value of our terminal point, we're going down to y is equal to two. So we have a change in y our change in y is equal to, going from nine to two, our change in y is negative seven. Similarly, our change in x we're going from x is equal to two, to x is equal to five. So our change in the horizontal direction is plus three. So our change in x where we can either think of this as the horizontal component of the vector. This is equal to positive three and as we've just seen we have drawn a right triangle and so we can use a Pythagorean theorem to figure out, to figure out the length of the hypotenuse and you might say wait, wait, a length of a side of a triangle can't have a negative value and that's why these squareds are valuable because it doesn't matter if you're taking a negative seven squared or a positive seven squared, you're going to get a positive value here and if you really just view this as a triangle, all you care about is the length of this side right over here or it's the magnitude of this side or the absolute value of it which is just going to be positive seven and so we can say this is going to be equal to the magnitude of our vector is going to be equal to so three squared is nine, nine, and then negative seven squared is positive 49. So plus 49 or once again you could view this as our change in y squared which is negative seven squared or you could say, well just look, the absolute value, the length of the side, we don't wanna think of a side as having a negative value, the negative really just says, hey we're going from the top to the bottom it gives us our direction, but if we just say the length of it it's seven, well you're there if you just use a Pythagorean theorem. Seven squared would also be 49 and so either way you get the magnitude of our vector is equal to the square root of nine plus 49 is going to be 57, I don't think I can simplify this radical too much, no that's it. So the magnitude of this vector is the square root of 57.