The cross product (or vector product) of the two three-dimensional real vectors \(\mathbf{a}\) and \(\mathbf{b}\), written \(\mathbf{a} \times \mathbf{b}\) or \(\mathbf{a} \wedge \mathbf{b}\), is another three-dimensional vector. It is defined as follows: \[\begin{pmatrix}a_1 \\ a_2 \\ a_3\end{pmatrix} \times \begin{pmatrix}b_1 \\ b_2 \\ b_3\end{pmatrix} = \begin{pmatrix}a_2 b_3 - a_3 b_2 \\ a_3 b_1 - a_1 b_3 \\ a_1 b_2 - a_2 b_1 \end{pmatrix}.\] It can also be calculated by expanding the “determinant” \[\begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k}\\ a_1 & a_2 & a_3\\ b_1 & b_2 & b_3 \end{vmatrix}.\]
This is a vector (hence the name ‘vector’ product). The definition does not easily extend to vectors \(\mathbf{a}\) and \(\mathbf{b}\) with \(n\) components for \(n\ne3\). (Compare this to the dot product, which outputs a scalar and which does have a simple generalisation.)
The cross product combines information about the lengths of \(\mathbf{a}\) and \(\mathbf{b}\) and their directions. Specifically, \[\mathbf{a} \times \mathbf{b} = |\mathbf{a}|\, |\mathbf{b}| \sin \theta \,\hat{\mathbf{n}}\] where \(|\mathbf{a}|\) and \(|\mathbf{b}|\) are the lengths (magnitudes) of \(\mathbf{a}\) and \(\mathbf{b}\), \(\theta\) is the angle between the two vectors (with \(0\le\theta\le\pi\)), and \(\hat{\mathbf{n}}\) is a unit vector perpendicular to both \(\mathbf{a}\) and \(\mathbf{b}\). The vector \(\hat{\mathbf{n}}\) is chosen so that \(\mathbf{a}\), \(\mathbf{b}\) and \(\hat{\mathbf{n}}\) satisfy the right-hand rule: if you use your right hand, point your first (index) finger in the direction of \(\mathbf{a}\) and your second (middle) finger in the direction of \(\mathbf{b}\), then your thumb, when lifted, will point in the direction of \(\hat{\mathbf{n}}\) and hence in the direction of \(\mathbf{a}\times\mathbf{b}\).
It follows that for non-zero vectors \(\mathbf{a}\) and \(\mathbf{b}\), the cross product is zero exactly when \(\mathbf{a}\) and \(\mathbf{b}\) are parallel to one another.
Any two vectors \(\mathbf{a}\) and \(\mathbf{b}\) define a parallelogram and a triangle. The area of the parallelogram is given by \(|\mathbf{a}\times\mathbf{b}|\), and the area of the triangle is half of this.
For every choice of \(\mathbf{a}\), \(\mathbf{b}\) and \(\mathbf{c}\) and every scalar \(\lambda\), we have \[\begin{align*} \mathbf{a} \times \mathbf{b} &= -\mathbf{b} \times \mathbf{a} && \text{(anti-commutativity)}\\ \mathbf{a} \times (\mathbf{b} + \mathbf{c}) &= \mathbf{a} \times \mathbf{b} + \mathbf{a} \times \mathbf{c} && \text{(distributivity over addition)}\\ (\lambda \mathbf{a}) \times \mathbf{b} &= \lambda (\mathbf{a} \times \mathbf{b}) = \mathbf{a} \times (\lambda \mathbf{b}). \end{align*}\]
