# Way Too Many Crossproducts!

Geometry Level 3

Suppose $$\vec{p} , \vec{q}, \text{ and } \vec{r}$$ are three mutually perpendicular unit vectors.

Vector $$\vec{u}$$ satisfies the equation $\vec{p}\times((\vec{u} - \vec{q})\times\vec{p}) \hspace{.15cm} + \hspace{.15cm} \vec{q}\times((\vec{u} - \vec{r})\times\vec{q}) \hspace{.15cm} + \hspace{.15cm}\vec{r}\times((\vec{u} - \vec{p})\times\vec{r}) = 0$

What is $$\vec{u}$$ in terms of $$\vec{p} , \vec{q}, \text{ and } \vec{r}$$?

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