The Tetrahedron Problem

Geometry Level 5

\( \vec{a},\vec{b}\) and \(\vec{c}\) are three non coplanar unit vectors inclined at an angle \(\alpha (\leq \dfrac{\pi}{2} ) \) to each other . If volume of the tetrahedron formed by these vectors is \(\dfrac{1}{\sqrt{360}}\). Then find the value of

\[ 3{\cos}^{2} \alpha -2\cos^{3} \alpha. \]

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