Cross Product

The cross product $\mathbf{a} \times \mathbf{b}$ is defined only in $\mathbb{R}^3$ and \mathbb{R}^ and produces a vector (unlike the Dot Product which produces a scalar):

Geometric interpretation:

• The result is perpendicular to both $\mathbf{a}$ and $\mathbf{b}$ (right-hand rule for direction)
• Its magnitude is the area of the parallelogram spanned by $\mathbf{a}$ and $\mathbf{b}$: $|\mathbf{a} \times \mathbf{b}| = |\mathbf{a}||\mathbf{b}|\sin\theta$

Key properties:

• Anti-commutative: $\mathbf{a} \times \mathbf{b} = -(\mathbf{b} \times \mathbf{a})$
• $\mathbf{a} \times \mathbf{b} = \mathbf{0}$ iff $\mathbf{a}$ and $\mathbf{b}$ are parallel
• Not associative: $(\mathbf{a} \times \mathbf{b}) \times \mathbf{c} \neq \mathbf{a} \times (\mathbf{b} \times \mathbf{c})$ in general
• The scalar triple product $\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})$ gives the volume of the parallelepiped

ML relevance: Less directly used than the dot product, but appears in 3D geometry, computer vision (surface normals), robotics, and physics simulations.

See also: Dot Product , Norms and Distance Metrics