# 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. 