Proving that cross product squared is an inner product determinant

Question

Prove that: $\left \| \vec{v}\times \vec{w}\right \|^2=det\begin{pmatrix} <\vec{v},\vec{v}> & <\vec{v},\vec{w}>\\ <\vec{w},\vec{v}> & <\vec{w},\vec{w}> \end{pmatrix}$

What is the relation between cross and inner product so that I can conclude the above equality?

Answer 1

I think I figured it out

$\left \| \vec{v}\times \vec{w} \right \|^2=\left \| \vec{v} \right \|^2\left \| \vec{w} \right \|^2-<\vec{v},\vec{w}>^2=\left \| \vec{v} \right \|^2\left \| \vec{w} \right \|^2-<\vec{v},\vec{w}><\vec{w},\vec{v}>=<\vec{v},\vec{v}><\vec{w},\vec{w}>-<\vec{v},\vec{w}><\vec{w},\vec{v}>=\begin{vmatrix} <\vec{v},\vec{v}> & <\vec{v},\vec{w}>\\ <\vec{w},\vec{v}> & <\vec{w},\vec{w}> \end{vmatrix}$