Cross product

The cross product combines two three-dimensional vectors to produce a vector perpendicular to both. Its magnitude equals the product of their lengths times the sine of the angle between them.

Determinant: A scalar or function computed from a square matrix that encodes properties such as invertibility and signed volume. A determinant computes the cross product's components from the input vectors.
Vector: A quantity with magnitude and direction, represented geometrically by an oriented segment or algebraically by components. The cross product takes two vectors as inputs and returns another vector.
Torque: A measure of the tendency of a force to produce rotation about an axis. Torque is the cross product of a position vector and a force.
Scalar triple product: A scalar formed by taking the dot product of one vector with the cross product of two others. It uses the cross product to measure signed volume rather than return a vector.
Right-hand rule: A convention that relates the directions of perpendicular vectors using the fingers and thumb of the right hand. It determines which of the two possible perpendicular directions the cross product takes.
Three-dimensional space: A space with three independent spatial coordinates. The familiar vector-valued cross product is defined for three-dimensional vectors.
Angular momentum: A conserved physical quantity describing rotational motion about a point or axis. For a particle, angular momentum is position crossed with momentum.
Quaternion: A number system extending the complex numbers with three imaginary units and noncommutative multiplication. Quaternion multiplication contains a cross-product term but remains defined beyond vector-only operations.
Parallelogram: A quadrilateral whose opposite sides are parallel and equal in length. The cross product's magnitude equals the area of the parallelogram spanned by its inputs.
Linear independence: A property of vectors where none can be expressed as a linear combination of the others. Parallel inputs are dependent and produce a zero cross product.

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