The cross product is a fundamental operation on vectors. It acts on vectors in three dimensions and results in another vector in three dimensions which is perpendicular to both of the other vectors!

Let \(\hat{u}\) and \(\hat{v}\) be unit vectors and \(\vec{w}\) be a vector such that \(\vec{w}+(\vec{w}\ \times \hat{u})\) \(=\) \(\hat{v}\).

The angle **in degrees** between \(\hat{u}\) and \(\hat{v}\) such that \(|(\hat{u} \times \hat{v}) \cdot \vec{w}|\) is maximized is \(\theta\) and the maximum value of \(|(\hat{u} \times \hat{v}) \cdot \vec{w}|\) is \(M\). Find the value of \(\theta + M\).

\(Image\) \(Credit :\) \(Wikipedia\)

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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