Cross product

The cross product, or vector product, is a type of vector multiplication in the Euclidean spaces, and is widely used in many areas of mathematics and physics. In $\mathbb {R} ^{3}$ there is another type of multiplication called the dot product ( or scalar product), but it is only defined and makes sense in general for this particular vector space. 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 = (A₁, ... ,A_n) and B = (B₁, ... ,B_n) in $\mathbb {R} ^{n}$ with $1\leq n\leq 3$, 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.

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