Suppose we have an inner product space V, with inner product $<x,y>$. In this space, we have two nonzero vectors u and v.
I am trying to find an arbitrary, real $a$ for which the following two vectors are orthogonal: $av-u$ and $v$.
I know two vectors are orthogonal when their inner product equals zero. When I take the inner product of these two vectors, I get $(av-u)v = 0$ . Is it correct that this means I should solve $(av-u)=0$?
And would $a$, therefore, be $u/v$?
Am I doing this correctly? thanks!
Best Answer
Just expand the inner product as follows (assuming $v \neq 0$, if it is, then the dot product is always zero and $v$ is orthogonal to every vector.) $$ <av-u|v> = a<v|v> - <u|v> = a||v||^2 - <u|v> = 0 \implies a = \frac{<u|v>}{||v||^2} $$