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:

Magnitude of the Cross Product

Magnitude of the Cross Product

The cross product can be expressed by the determinant:

Cross Product as a Determinant

Examples

Calculate the cross product of the vectors Vector U = (1, 2, 3) and Vector V = (−1, 1, 2).

Cross Product Example

Cross Product Solution


Find the cross product of the vectors Vector U and Vector V and check that the resultant vector is orthogonal to Vector U and Vector V.

Cross Product Example

Cross Product Solution

Cross Product Solution

The cross product of Cross Product is orthogonal to the vectors Vector U and Vector V.


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.

Cross Product of Two Vectors

Area of a Parallelogram


Example

Find the area of the parallelogram which is formed by the vectors Vector Uand Vector V.

Cross Product Example

Cross Product Solution


Area of a Triangle

Area of a Triangle

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).

Vectors Example

Cross Product Operations

Cross Product Calculation

Vector W

Vector Magnitude

Area Solution


Cross Product Properties

1. Anticommutative.

Vector U x Vector V = −Vector V x Vector U

2. Compatible with scalar multiplication.

k (Vector U x Vector V) = (kVector U) x Vector V = Vector U x (kVector V)

3. Distributive over addition.

Vector U x (Vector V + Vector W ) = Vector U x Vector V + Vector U x Vector W ·

4. The cross product of two parallel vectors is equal to the zero vector.

Vector U modo normal se ignoro de paralelo Vector V flechas Vector U x Vector V = Zero Vector

5. The cross product Vector U x Vector V is perpendicular to Vector U and Vector V.