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we've already seen that a vector is defined by both its magnitude and its direction and 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 coordinate plane and I want to figure out its magnitude and I encourage you to pause the video and see if you can configure it out well the magnitude of a vector is 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 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 and 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 9 and we are to get to the Y value of our terminal point we're going down to Y is equal to 2 so we have a change in Y our change in Y is equal to going from 9 to 2 our change in Y is negative 7 similarly our change in X we're going from X is equal to 2 to X is equal to 5 so our change in the horizontal direction is plus 3 so our change in X or we could even think of this as the horizontal component of the vector this is equal to positive 3 and as we've just seen we have drawn a right triangle and so we could 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 an egg value and that's why these Squared's are valuable because it doesn't matter if you're taking a negative seven squared or a positive seven square 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 that'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 3 squared is 9 9 and then negative 7 squared is positive 49 so plus 49 well once again you could view this as our change in Y squared which is negative 7 squared or you could say well I'll just look the absolute value the length of the side we don't want to 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 7 when you there if you just use the Pythagorean theorem 7 squared would also be 49 and so either way you get the magnitude of our vector is equal to the square root of 9 plus 49 is going to be 50 57 now I don't think I can simplify this radical too much that's that's it so the magnitude of this of this vector is the square root of 57