Cross product

The cross product, or vector product, is a type of vector multiplication in $\mathbb {R} ^{3}$, and is widely used in many areas of mathematics and physics. In general Euclidean spaces there is another type of multiplication called the dot product ( or scalar product). Both the dot product and the cross product are widely used in in the study of optics, mechanics, electromagnetism, and gravitational fields, for example.

Definition

Given two vectors, A = (A_x,A_y,A_z) and B = (B_x,B_y,B_z) in $\mathbb {R} ^{3}$, the cross product is defined as the vector product of the magnitude of A, the magnitude of B, the sine of the smaller angle between them, and a unit vector (a_N) that is perpendicular (or normal to) the plane containing vectors A and B and which follows the right-hand rule (see below).

A x B = a_N |A||B|sinθ_AB

where $|\mathbf {A} |=(\mathbf {A} \cdot \mathbf {A} )^{1/2}$ and $|\mathbf {B} |=(\mathbf {B} \cdot \mathbf {B} )^{1/2}$ are, respectively, the magnitudes of A and B. See dot product for the evaluation of this equation.

Reversing the order of the vectors A and B results in a unit vector in the opposite direction, meaning that the cross product is not commutative, and thus:

B x A = -(A x B)

The cross product of any vector with itself (or another parallel vector) is zero because the sin(0) = 0.

A x A = 0

Another formulation

Rather than determining the angle and perpendicular unit vector to solve the cross product, the form below is often used to solve the cross product in $\mathbb {R} ^{3}$.

A x B = (A_yB_z - A_zB_y)a_x + (A_zB_x - A_xB_z)a_y + (A_xB_y - A_yB_x)a_z,

where a_x, a_y and a_z are the orthonormal bases on which A and B have been defined. The above formula can be written more concisely in the following form:

$\mathbf {A} \times \mathbf {B} =\left|{\begin{array}{ccc}\mathbf {a} _{x}&\mathbf {a} _{y}&\mathbf {a} _{z}\\A_{x}&A_{y}&A_{z}\\B_{x}&B_{y}&B_{z}\end{array}}\right|,$

where $\left|\cdot \right|$ denotes the determinant of a matrix.

The right hand rule

The diagram below illustrates the direction of A x B, which follows the right hand rule. If one points the fingers of the right hand towards the head of vector A (with the wrist at the origin), then curls them towards the direction of B, the extended thumb will point in the direction of A x B.

Vector Diagram illustrating the direction of A x B