Cross Product
Table of Contents
For two vectors \(\small \vec{a}\) and \(\small \vec{b}\) in \(\small \mathbb{R}^3\) (real coordinate spaces, the cross product is defined as
$$ \vec{a} \times \vec{b} = \begin{vmatrix} \vec{i} & \vec{j} & \vec{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix}$$
Where
- The right hand side is a determinant of a 3x3 matrix,
- \(\small a_i\) and \(\small b_i\) are the \(\small i\)-th components of \(\small \vec{a}\) and \(\small \vec{b}\), respectively, and
- \(\small \vec{i}\), \(\small \vec{j}\), and \(\small \vec{k}\) are the standard basis vectors of \(\small \mathbb{R}^3\).
Interpretation #
The result of a cross product is a vector that is perpendicular to both \(\small \vec{a}\) and \(\small \vec{b}\).
The magnitude of the cross product is equal to the area of the parallelogram formed by \(\small \vec{a}\) and \(\small \vec{b}\), and can be calculated:
$$ |a \times b| = |a| |b| \sin \theta $$
Where \(\small \theta\) is the angle between \(\small \vec{a}\) and \(\small \vec{b}\).
The direction of the cross product can be determined by the right-hand rule. If you point your right hand in the direction of \(\small \vec{a}\) and curl your fingers in the direction of \(\small \vec{b}\), your thumb will point in the direction of \(\small \vec{a} \times \vec{b}\).
Properties #
- The cross product is anti-commutative: \(\small \vec{a} \times \vec{b} = - \vec{b} \times \vec{a}\)
- The cross product is distributive: \(\small \vec{a} \times (\vec{b} + \vec{c}) = \vec{a} \times \vec{b} + \vec{a} \times \vec{c}\)