Finding a reflection that exchanges two unit vectors

Question

Let $v$ and $u$ be two vectors in an Euclidean space, that have same length:$\|v\| = \|u\|$, but are not on a same line. Find a reflection w that commute/switch between $u$ and $v$.

Answer 1

This visual aid should give you a better idea of how to achieve what you're trying to achieve. The vector perpendicular to the reflection plane is seen to be

$ \frac{u - v}{|u -v|} $ let's call this vector $\hat{r}$

Then we can write down the reflection of any vector $x$ as:

$x \rightarrow x - 2((x - \frac{u + v}{2}) \cdot\hat{r} ) \hat{r} $ , so substituting we have: $x \rightarrow x - 2((x - \frac{u + v}{2}) \cdot \frac{u - v}{|u -v|} ) \frac{u - v}{u-v}$

now if we substitute for $u$ for $x$ then we have

$u \rightarrow u - 2 (\frac{u-v}{2} \cdot \frac{u - v}{|u-v|} ) \frac{u-v}{|u-v|} = u - (u - v) = v$

And similarly it follow for v -> u. Or alternatively you can notice that it is antisymmetric in u and v.