How to Calculate the Projection W of X Along Y

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Calculating the projection of W of X along Y wi9ll require you to use a standard plane. Calculate the projection of W of X along Y with help from an experienced mathematics educator in this free video clip.

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Video Transcript

Hello, my name is Walter Unglaub, and this is how to calculate the projection W of X along Y. So in my standard X, Y plane, I have here a vector V which has some magnitude and it's also separated from the X axis by this angle theta. When I do a projection along each of these axes, I end up with two orthogonal vectors, V, X and V, Y such that I can rewrite my expression for this general vector V as the magnitude V, X times the univector I plus magnitude V, Y times the unit vector J. Now, if I want to compute what these components are, I have to first understand what the magnitude of the original vector is and that's given by this modulus of V and hence by using Pythagorean Theorem and a bit of geometry, I have V, X is equal to the magnitude of V, the hypotenuse times cosine of the angle theta. Likewise for V, Y I have the magnitude of the vector times sine of theta. Now what we are interested in is determining what the projection W is of the X axis on the Y axis. So one way of thinking that in terms of this framework is if our V, X vector was infinite in magnitude and lied purely of course along the X axis and then I want to compute what is the inner product of that with the Y axis. So to do this, I have W which is going to be the projection onto the Y axis since that's what we are interested in. So the W is going to actually be V, Y and this is simply going to be the magnitude of V which in this case is V, X times sine of the angle theta. But you'll notice that if my vector lies purely along the X axis, that the angle theta is going to be equal to 0, 0 degrees or alternatively 0 radians. Sine of 0 is equal to 0. Therefore, my projection, the shadow of this X axis, this vector that lies all along the X axis onto the Y axis because they are separated by 90 degrees, we call the orthogonal and we calculate the projection to simply be 0 because the angle between the vector that lies along the X axis and the X axis is 0. So this is our projection W. My name is Walter Unglaub and this is how to calculate the projection W of X along the Y axis.

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