Ratio between inner products on a vector space 
Let $V$ be a vector space and let$ \langle,\rangle_1$  and $\ \langle,\rangle_2$ be inner products in $V$ s.t. $\langle,\rangle_1=0  \iff \langle,\rangle_2=0$.
Prove $ \langle v,w\rangle_1=c\langle v,w\rangle_2$ for every $v,w \in V$.

I've been struggling coming up with a solution for this. This is what I have so far:
Let $U$ be a subspace of $V$. We know $\ U \oplus U^{\bot}=V$ for $\langle,\rangle_1$  and $\langle,\rangle_2$.
Let $\ B=\{u_1,...u_n\}$ be an orthonormal basis for $U$ and $\ C=\{v_1,...v_k\}$ an orthonormal basis for $\ U^{\bot}$ under $\langle,\rangle_1$.
$\ B'=\{\frac{u_1}{||u_1||},...,\frac{u_n}{||u_n||}\}$ is an orthonormal basis for $U$ and $\ C'=\{\frac{v_1}{||v_1||},...,\frac{v_k}{||v_k||}\}$ an orthonormal basis for $\ U^{\bot}$ under $\langle,\rangle_2$.
Take$\ u \in B $ and $\ v \in C$. If I prove for base vectors the argument is valid for any vector in the subspace.
I'd like to find the ratio using the norms of the bases B' and C', I'm not sure how to get at it and if this is the right direction.

You do not need to assume finite dimensionality of $V$. The result you are trying to prove holds in any inner product space, real or complex, of any (possibly infinite) dimension. Summoning an orthonormal basis just complicates the problem. You should start by figuring out what the number $c$ must be. The answer is $c=\left\langle{u,u}\right\rangle_1/\left\langle{u,u}\right\rangle_2$, where $u$ is any nonzero vector in $V$ (If $V$ is the zero space, the statement to be proved is trivially true, so we can assume $V$ has nonzero vectors). For this definition of $c$ to make sense however, you have to show that it does not depend on the choice of $u$. In other words, you have to show that $$\frac{\left\langle{u,u}\right\rangle_1}{\left\langle{u,u}\right\rangle_2}=\frac{\left\langle{v,v}\right\rangle_1}{\left\langle{v,v}\right\rangle_2}$$ holds for all nonzero $u,v\in V$. Once you've done that, showing that $c$ is the desired ratio is straightforward.
