Eroxl's Notes
Cross Product

The cross product is a geometric operation that takes 2 vectors in 3 dimensional Euclidean space and returns a vector that is perpendicular to the plane between both input vectors.

The cross product is only defined in 3 dimensional Euclidean space and is usually represented using a symbol. The cross product is anticommutative meaning .

Definition

Geometric

The cross product can be calculated geometrically as follows:

Where is the angle between and . Additionally the direction of the cross product can be calculated using the right hand rule.

Numerically

The cross product can be defined with the notation of determinants as follows:

Identities

Applications

Area of a Parallelogram

To calculate the area of the parallelogram formed between and is given by the equation:

Examples

Simple Cross Product 1

Find the cross product where and .

Simple Cross Product 2

Find the cross product where and .

Orthogonal Vector

Find two unit vectors orthogonal to and .

Unit Vector Orthogonal to Plane

Find a unit vector with positive first coordinate that is orthogonal to the plane through the points , , and .