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**1) Let U be a plane through the origin in R^3 with a nonzero normal vector n=[a b c]^T. Find the projection matrix of X=[x1 x2 x3]^T onto U.**

I got this question from my linear algebra test today and I am dying on it.

I tried something out but ended up with a terribly ugly result in which I have no confidence of it being right.

My method:

Since (projection of X onto n) gives the perpendicular (closeest) distance from X to the plane U, I have the following inequality: (in orthongonal complement of U)

**(projection of X onto n) = X - (projection of X onto U)**(<-is this right?)

and then solve for (projection of X onto U) for which I can obtain the induced matrix by factoring the matrix [x1 x2 x3]^T out

and this ends up with some ugly calculations (this question only worth 5 marks, how can I take that long?)

Is there a flaw in this thinking? Is it right?

I seriously think I have missed something...Is there a very easy method to do this question? Can someone teach me? I can't sleep without it.

Thanks a lot!