[Math] Determining an unknown vector from its cross and dot product with known vector

cross productvectors

Let $\vec{k}$, $\vec{v}$, and $\vec{u}$ be vectors, where $\vec{u}$ is unknown and $\vec{k}$ and $\vec{v}$ are known vectors. Given:

  1. $\vec{u}\cdot\vec{k}=c$
  2. $\vec{u} \times \vec{k}= \vec{v}$

From this relations, how can i determine the vector $u$? I tried to construct orthogonal coordinate system from $(\vec{k}, \vec{v},\vec{k}\times \vec{v})$ but i couldn't proceed from there. Any idea?

Best Answer

From

$$ \vec k\times(\vec u\times\vec k)=\vec u(\vec k\cdot\vec k)-\vec k(\vec k\cdot\vec u) $$

you get

$$ \vec u=\frac{\vec k\times(\vec u\times\vec k)+\vec k(\vec k\cdot\vec u)}{\vec k\cdot\vec k}=\frac{\vec k\times\vec v+c\vec k}{\vec k\cdot\vec k}\;. $$

(Of course you need $\vec k\ne\vec0$.)

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