Magnittude of the Cross or Vector Product of Two Vectors

The magnitude of the cross product of vectors  
\[\vec{a}\]
  and  
\[\vec{b}\]
  is  
\[\| \vec{a} \times \vec{b} \| = \| \vec{a} \| \| \vec{b} \| sin \theta\]
  where  
\[\theta\]
  is the angle between  
\[\vec{a}\]
  and  
\[\vec{b}\]
.
\[\begin{equation} \begin{aligned} \| \vec{a} \|^2 \| \vec{b} \|^2 sin^2 \theta &= \| \vec{a} \|^2 \| \vec{b} \|^2 (1-cos^2 \theta ) \\ &= \| \vec{a} \|^2 \| \vec{b} \|^2 -\| \vec{a} \|^2 \| \vec{b} \|^2 cos^2 \theta \\ &=\| \vec{a} \|^2 \| \vec{b} \|^2 - (\vec{a} \cdot \vec{b})^2 \\ &= (a_1^2+a_2^2+a_3^2)(b_1^2+b_2^2+b_3^2)- (a_1b_1+a_2b_2+a_3b_3)^2 \\ &= (a_2b_3-a_3b_2)^2+(a_3b_1-a_1b_3)^2+(a_1b_2-a_2b_1)^2 \\ &= \| \vec{a} \times \vec{b} \|^2\end{aligned} \end{equation} \]