Vectors / HL

Vector Product and Area

Use the cross product to find a vector perpendicular to two vectors and to calculate areas.

Key ideas

\[|\mathbf a\times\mathbf b|=|\mathbf a||\mathbf b|\sin\theta\]

\(\mathbf a\times\mathbf b\) is perpendicular to both \(\mathbf a\) and \(\mathbf b\).
\(|\mathbf a\times\mathbf b|\) is the area of the parallelogram formed by \(\mathbf a\) and \(\mathbf b\).
The triangle area is \(\frac12|\mathbf a\times\mathbf b|\).

Tip-to-tail addition: the resultant vector goes from the start of the first vector to the end of the second.

If \(\mathbf a=(3,0,0)\) and \(\mathbf b=(0,4,0)\), then the parallelogram area is \(3\times4=12\), so the triangle area is \(6\).