# Maximized Parallelepiped

**Algebra**Level 4

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 \(\theta + M\).

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