# Cross Product

The result of the cross-product of a two vectors is another vector. Given two vectors and , their cross-product can be written as:

or, using determinants:

which is derived from:

## Properties

• The length of the resulting vector representing the cross-product of two vectors is:

where the vertical bars represent the scalar value of the vector.

• The length of the cross-product of two vectors is equal to the area of the parallelogram determined by the two vectors.

• Multiplication by scalars is commutative. For example, suppose that is a scalar and that is the cross product of respectively , in that case we have that:

• The cross-product is distributive.

• The volume of the parallelepiped determined by the vectors , and is equal to the magnitude of their scalar triple product.
• The vector triple product of the vectors , and is given by:

• Two vectors are parallel if and only if their cross product is the zero vector.

# Determining Triangle Area in Three Dimensions

One solution to find the area of a triangle is to determine its height and then proceed to the multiply the height by the base and divide by two. However, that may be a difficult task and instead, by using the parallelogram property of vectors, one can calculate the area of the parallelogram and then divide the area by two.