Vector with equal projection onto two other vectors

Question

Consider two vectors $\vec{u}$ and $\vec{v}$ such that $|\vec{u}|\neq |\vec{v}| \neq 1$.

Consider a third vector $\vec{a}$ such that

$$\vec{a}.\vec{u}=\vec{a}.\vec{v}\neq 0$$

The question is to find $\vec{a}$.

Answer 1

Let $u_i$ and $v_i$ be the $i^{th}$ entry of $u,v$, respectively. This means that $$\sum u_i a_i = \sum v_ia_i $$ or $$\sum_{i=1}^N (u_i-v_i) a_i = 0 $$ We have many choices here: EIther $a_i$'s are zero, but that's trivial. Instead we could pick $a_i = \frac{1}{u_i-v_i}$ if $u_i \neq v_i$ and a corresponding $a_j = -\frac{1}{u_j-v_j}$ if $u_j \neq v_j$. You do that repeatedly until the terms sum to zero. If one term is left out, you set that $a_i$ to zero.