Maximized Parallelepiped

Algebra Level 3

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

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

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