Cross product ↩

shorthands: {}
aliases: [Vector product, cross product]
created: 2021-12-13 16:44:48
modified: 2022-06-24 02:27:59

Definition

The cross product is only defined in three dimensional space and is denoted . The cross product is defined as a vector that is perpendicular (orthogonal) to bot and with a direction given by the right-hand rule and a magnitude equal to the area of the parallelogram that the vectors span.

Where is the angle between and , is the Euclidean norm and is the unit vector that is perpendicular to both vector, in the direction of the right-hand rule.

Special cases:

• If and are parallel, then

Calculation

Here form an orthonormal basis on the Euclidean vector space.

Using matrix determinant

We can calculate it using matrix determinant:

Direct calculation

Using the levi-civita symbol

We can use the Levi-Civita symbol to express cross product: