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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!

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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 \(\theta + M\).

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

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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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Consider the regular octagon centered at the origin as shown above. Eight unit vectors are drawn from the center of the octagon to each of its vertices and labeled in the figure. For each pair of distinct unit vectors \(\vec{u}_i, \vec{u}_j\) with \(i < j\), their cross product is computed. What is the sum of all of these cross products?

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