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Anderson Video - Vector Components

Professor Anderson
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>> Hello, class. Professor Anderson here. Let's talk a little bit about vector components, how we can break these vectors into their x and y components. And let's think about our x and y coordinate space. So here is x, here is y, and now let's draw a vector in that space, okay. There is my Vector A. That is going up to a particular point here labeled by x and y, okay. And we know exactly what x and y are. x is, how far did I go across in x? y is, how far up did I go in y? And so this side is A sub x and this side is A sub y, okay. And both of those are in fact vectors because if I add Ax and Ay, I get my resultant Vector A. A is just Ax plus Ay. But we also know a little bit about polar coordinates now. So if that is angle theta and the hypotenuse is the magnitude of the Vector A, which is just A in this case without the arrow, what can we say about these components? A sub x is the magnitude of A sub x. How long is this? And that is the hypotenuse of the triangle, which is A times the cosine theta. Okay, A sub x is equal to A cosine theta. A sub y is this side of the triangle, which is, of course, just A sine theta. And the magnitude A is just the square root (Ax squared plus Ay squared), and that angle theta you can write in terms of any sides you want, but let's take the tangent. And therefore, the tangent of theta would give us Ay over Ax, okay. So this is how you can define the parameters of this vector and break it out into components. Now, we need to describe this in terms of unit vectors next.