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, Cesareo Apr 20 at 7:16

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

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.

  • $\begingroup$ Indeed, "some other vector" can be $v$ unless it's parallel to $u$. $\endgroup$ – J.G. Apr 19 at 21:23
  • 1
    $\begingroup$ yeah exactly right $\endgroup$ – George Dewhirst Apr 19 at 21:23

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