The cross product or vector product is a way of multiplying two vectors together to produce another vector that is orthogonal to both of the original vectors.

The formula for the cross product is:

where \(\mathbf{n}\) is a unit vector (vector of length \(1\)) that is orthogonal (at right angles) to both \(\mathbf{a}\) and \(\mathbf{b}\).

For example, if two vectors have lengths \(4\) and \(5\) and the angle between them is \(30^\circ\), and \(\mathbf{n}\) is a unit vector that is orthogonal to both \(\mathbf{a}\) and \(\mathbf{b}\), then

\(\mathbf{a} \times \mathbf{b} = (4 \times 5 \times \sin(30^\circ))\mathbf{n} = 10 \mathbf{n}.\)