How to find the cross product

Since cross product or vector product is a binary operation between two vectors in three-dimensional vector space, it is useful to know how to find the cross product. The cross product of two vectors leads to another vector perpendicular to the plane containing the first two vectors. In general, the cross product or vector product is symbolized by the multiplication sign, but the mathematical operation is more advanced than simple algebraic multiplication.

The cross product of vectors $\ bar {A}$ and $\ Barb}$ is described as $\ bar {A} \ times \ bar {B}$ and creates another vector $\ bar {C}$ that is perpendicular to both $\ bar {A}$ and $\ Barb}$ .

where θ is the angle measured by $\ bar {A}$ to $\ Barb}$ and η is the unit vector in the direction perpendicular to the plane containing both $\ bar {A}$ and $\ Barb}$ .

Geometrically, the amount of the cross product of two vectors is equal to the area of a parallelogram with $\ bar {A}$ and $\ Barb}$ as adjacent pages. Vectors $\ bar {A}$ , $\ Barb}$ and $\ bar {C}$ for a right-handed system as follows:

The cross product has the following algebraic properties.

The following results also apply to the cross product.

How to find the cross product

Vectors are often specified as components within a coordinate system. In this form, it is convenient to use determinants to calculate the cross product.