I was trying to prove that proj_v(u)=\frac{v·u}{||v||^2} v, and I was getting close, but then a friend spoiled the fun of completing the proof by giving me what he called a "hint". Blurting out a key part in a proof isn't a "hint". Needless to say, after this, I immediately knew what to do to complete the proof, but as I looked back at my original approach, I can't help but wonder what was wrong with it. I tried it, but with it I got answers like the zero vector or even an infinite number of answers, and I don't know why.
What I did was I defined the projection of u onto v to be the vector x such that (u- x)·v=0 (because of the orthogonality). If I break these vectors down into their component forms, with the components of x $\langle$ x$_1$, x$_2\rangle$ being treated as variables, (and the components of u and v as constants) then we get an infinite number of answers, since we have 2 variables. Why does this approach to the problem give me an infinite number of answers, when there is clearly one unique solution? Thank you.