Way Too Many Crossproducts!

Geometry Level 3

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

Vector u\vec{u} satisfies the equation p×((uq)×p)+q×((ur)×q)+r×((up)×r)=0\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 u\vec{u} in terms of p,q, and r\vec{p} , \vec{q}, \text{ and } \vec{r}?

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