GEOMETRY

# 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.