Cross Product

Overview

The cross-product is a mathematical operation you can perform on two vectors in 3D space. The cross-product produces a vector which is orthogonal to both of the input vectors, which means it is also normal to the plane containing the two input vectors.

Mathematical Symbol

The cross product of vectors \(\vec{a}\) and \(\vec{b}\) is written as:

$$ \vec{a} \times \vec{b} $$

Defining Equation

The cross product is defined by the formula:

$$ \vec{a} \times \vec{b} = ||\vec{a}|| ||\vec{b}|| \, sin (\theta) \, \vec{n} $$

where:
\(\theta\) is the angle between the vectors
\(\vec{n}\) is the vector which is normal to both \(\vec{a}\) and \(\vec{b}\)

Cross Product Properties

The cross product of two vector always produces a vector which:

• Is orthogonal to both input vectors (i.e. normal to the plane containing the input vectors).
• Has a direction which is determined by the right-hand rule.
• Has a magnitude which is equal to the area of the parallelogram formed by the two input vectors.