I have a question:
Find the projection of $v = <1,2,1>$ on $span(<3,1,2>,<1,0,1>)$ in $R^3$
calling the vectors in the span a and b $$proj_w V = \frac{V \cdot a }{a^2} \vec{a} + \frac{V \cdot b }{b^2} \vec{b}$$
I've checked my work a few times.. $<\frac{5}{2} , \frac{1}{2}, 2>$ which isn't right via back of the book. I also think you can check via $V - proj_w V$ being orthogonal to all the vectors in the set.. which it is not.
What am I missing here?
A projection $P$ must satisfy $P^2=P$. That is whatever the vector which is on the projection space remains invariant under projection. Obviously $v=a$ or $v=b$ are counterexamples. To find the projection of $v$ on subspace, one should find an orthogonal basis of the space and then apply the formula you did.