Cross Product
The cross product of two vectors is another perpendicular vector to the two vectors.
The direction of the resultant vector can be determined by the right-hand rule. The thumb (u) and index finger (v) held perpendicularly to one another represent the vectors and the middle finger held perpendicularly to the index and thumb indicates the direction of the cross vector.
The magnitude of the cross product is:
The cross product can be expressed by the determinant:
Examples
Calculate the cross product of the vectors 
= (1, 2, 3) and 
= (−1, 1, 2).
Find the cross product of the vectors 
and 
and check that the resultant vector is orthogonal to and .
The cross product of 
is orthogonal to the vectors and .
Area of a Parallelogram
Geometrically, the magnitude of the cross product of two vectors coincides with the area of the parallelogram whose sides are formed by those vectors.
Example
Find the area of the parallelogram which is formed by the vectors 
and 
.
Area of a Triangle
Example
Determine the area of the triangle whose vertices are the points A = (1, 1, 3), B = (2, −1, 5) and C = (−3, 3, 1).
Cross Product Properties
1. Anticommutative.
x = − x
2. Compatible with scalar multiplication.
k ( x ) = (k) x = x (k)
3. Distributive over addition.
x ( + 
) = x + x ·
4. The cross product of two parallel vectors is equal to the zero vector.
x = 
5. The cross product x is perpendicular to and .
