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I need help with the following exercise:

Given the vectors $u_1 = (2,-1,2), u_2 = (1,2,1), u_3 = (-2,3,3)$, what is the projection of $u_3$ onto the plane spanned by $u_1$ and $u_2$.

I'm not sure if I need to use the GS process to get an orthonormal basis for the plane and then project (I know how to do that) or if I should do like:

$(u_3)_{\| U} = \frac{\langle u_1 , u_3 \rangle}{\langle u_1 , u_1 \rangle} u_1 + ...$

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In order to apply the projection theorem, you do not need the vectors forming the basis of the plane to be orthogonal to each other. You do, however, need these vectors to form a basis. In your case, this simply means confirming that $u_1$ and $u_2$ are linearly independent.

Assuming this is true, you can project $u_3$ onto this plane exactly as you think you would.

Caveat: When you're performing this on a computer, you'll often find the algorithm to be much more numerically stable if these vectors are orthogonal to each other. If not, then you run the risk of dividing by a number very close to 0 and having your answers "blow up" even when you should theoretically be able to compute the correct answer as is.

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  • $\begingroup$ So if they are LI I'd just do $(u_3)_{\| U} = \frac{\langle u_1 , u_3 \rangle}{\langle u_1 , u_1 \rangle} u_1 + ...$ ? $\endgroup$ Jun 29 '15 at 21:17
  • $\begingroup$ @LorenzoPiccoliMódolo Yes, you would apply the projection theorem. $\endgroup$ Jun 29 '15 at 21:37

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