Editing Cross product (section)

== Applications ==
The cross product has applications in various contexts. For example, it is used in computational geometry, physics and engineering.
A non-exhaustive list of examples follows.

=== Computational geometry ===
The cross product appears in the calculation of the distance of two [[Skew lines#Distance|skew lines]] (lines not in the same plane) from each other in three-dimensional space.

The cross product can be used to calculate the normal for a triangle or polygon, an operation frequently performed in [[computer graphics]]. For example, the winding of a polygon (clockwise or anticlockwise) about a point within the polygon can be calculated by triangulating the polygon (like spoking a wheel) and summing the angles (between the spokes) using the cross product to keep track of the sign of each angle.

In [[computational geometry]] of [[Plane (geometry)|the plane]], the cross product is used to determine the sign of the [[acute angle]] defined by three points <math> p_1=(x_1,y_1), p_2=(x_2,y_2)</math> and <math> p_3=(x_3,y_3)</math>. It corresponds to the direction (upward or downward) of the cross product of the two coplanar [[vector (geometry)|vector]]s defined by the two pairs of points <math>(p_1, p_2)</math> and <math>(p_1, p_3)</math>. The sign of the acute angle is the sign of the expression
:<math> P = (x_2-x_1)(y_3-y_1)-(y_2-y_1)(x_3-x_1),</math>
which is the signed length of the cross product of the two vectors.
To use the cross product, simply extend the 2D vectors <math>p_1, p_2, p_3</math> to co-planar 3D vectors by setting <math>z_k=0</math> for each of them.

In the "right-handed" coordinate system,  if the result is 0, the points are [[collinear]]; if it is positive, the three points constitute a positive angle of rotation around <math> p_1</math> from <math> p_2</math> to <math> p_3</math>, otherwise a negative angle. From another point of view, the sign of <math>P</math> tells whether <math> p_3</math> lies to the left or to the right of line <math> p_1, p_2.</math>

The cross product is used in calculating the volume of a [[polyhedron]] such as a [[tetrahedron#Volume|tetrahedron]] or [[parallelepiped#Volume|parallelepiped]].

=== Angular momentum and torque ===
The [[angular momentum]] {{math|'''L'''}} of a particle about a given origin is defined as:

:  <math>\mathbf{L} = \mathbf{r} \times \mathbf{p},</math>

where {{math|'''r'''}} is the position vector of the particle relative to the origin, {{math|'''p'''}} is the linear momentum of the particle.

In the same way, the [[Moment (physics)|moment]] {{math|'''M'''}} of a force {{math|'''F'''<sub>B</sub>}} applied at point B around point A is given as:

:  <math> \mathbf{M}_\mathrm{A} = \mathbf{r}_\mathrm{AB} \times \mathbf{F}_\mathrm{B}\,</math>

In mechanics the ''moment of a force'' is also called ''[[torque]]'' and written as <math>\mathbf{\tau}</math>

Since position {{nowrap|{{math|'''r'''}},}} linear momentum {{math|'''p'''}} and force {{math|'''F'''}} are all ''true'' vectors, both the angular momentum {{math|'''L'''}} and the moment of a force {{math|'''M'''}} are ''pseudovectors'' or ''axial vectors''.

=== Rigid body ===
The cross product frequently appears in the description of rigid motions. Two points ''P''  and ''Q'' on a [[rigid body]] can be related by:

: <math>\mathbf{v}_P - \mathbf{v}_Q = \boldsymbol\omega \times \left( \mathbf{r}_P - \mathbf{r}_Q \right)\,</math>

where <math>\mathbf{r}</math> is the point's position, <math>\mathbf{v}</math> is its velocity and <math>\boldsymbol\omega</math> is the body's [[angular velocity]].

Since position <math>\mathbf{r}</math> and velocity <math>\mathbf{v}</math> are ''true'' vectors, the angular velocity <math>\boldsymbol\omega</math> is a ''pseudovector'' or ''axial vector''.

=== Lorentz force ===
{{See also|Lorentz force}}
The cross product is used to describe the [[Lorentz force]] experienced by a moving electric charge {{nowrap|{{math|''q<sub>e</sub>''}}:}}
: <math>\mathbf{F} = q_e \left( \mathbf{E}+ \mathbf{v} \times \mathbf{B} \right)</math>

Since velocity {{nowrap|{{math|'''v'''}},}} force {{math|'''F'''}} and electric field {{math|'''E'''}} are all ''true'' vectors, the magnetic field {{math|'''B'''}} is a ''pseudovector''.

=== Other ===
In [[vector calculus]], the cross product is used to define the formula for the [[vector operator]] [[Curl (mathematics)|curl]].

The trick of rewriting a cross product in terms of a matrix multiplication appears frequently in [[epipolar geometry|epipolar]] and multi-view geometry, in particular when deriving matching constraints.