Vector (Cross) Product

The vector (cross) product \(\vec a\times\vec b=|\vec a||\vec b|\sin\theta\,\hat n\), where \(\hat n\) is the unit vector perpendicular to both, chosen by the right-hand rule. In components:

\[\vec a\times\vec b=\begin{vmatrix}\hat i&\hat j&\hat k\\a_1&a_2&a_3\\b_1&b_2&b_3\end{vmatrix}\]

• \(\vec b\times\vec a=-\vec a\times\vec b\) (not commutative).
• \(\vec a\times\vec a=\vec0\); \(\vec a\parallel\vec b\iff\vec a\times\vec b=\vec0\).
• \(\hat i\times\hat j=\hat k\), \(\hat j\times\hat k=\hat i\), \(\hat k\times\hat i=\hat j\).
• Area of parallelogram with sides \(\vec a,\vec b\) = \(|\vec a\times\vec b|\); area of triangle = \(\frac12|\vec a\times\vec b|\).