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

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!

## closed as off-topic by user26857, Lee David Chung Lin, Chinnapparaj R, blub, CesareoApr 20 at 7:16

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• Since the question asks for all vectors, the solution would be $v+\mbox{span}(u)^{\bot}$. – Dog_69 Apr 19 at 21:37

## 1 Answer

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.

• Indeed, "some other vector" can be $v$ unless it's parallel to $u$. – J.G. Apr 19 at 21:23
• yeah exactly right – George Dewhirst Apr 19 at 21:23