Given: $u, v \in \mathbb{R}^3$ find all vectors $w$ such that $u \cdot v = w \cdot u$

Question

I have the following problem, but I do not know how to approach it.

Given: $u, v \in \mathbb{R}^3$ find all vectors $w$ such that $u \cdot v = w \cdot u$ (dot product)

Can anyone give me a hint on what should I do? Thanks in advance!

Answer 1

This is the same as $(w-v)\cdot u = 0$.

So find a vector orthogonal to $u$ then add $v$ to it to get $w$.

To get a vector orthogonal to $u$ you can do a cross product with $u$ and some other vector.