Vector

A vector is an element of a vector space.

An axial vector transforms as ${\mathbf{a}}^{\prime }=(\det Q)Q\mathbf{a}$ under an orthogonal matrix $Q$. It agrees with a polar vector under a proper rotation matrix and has the additional determinant sign under an orientation-reversing transformation. Angular momentum and angular velocity are axial vectors.

A polar vector transforms as ${\mathbf{v}}^{\prime }=Q\mathbf{v}$ under an orthogonal matrix $Q$, including an orientation-reversing transformation. Displacement, velocity and force are polar vectors. An axial vector instead transforms as ${\mathbf{a}}^{\prime }=(\det Q)Q\mathbf{a}$.

A Euclidean vector is a vector in a finite-dimensional real inner-product space, represented in Cartesian coordinates by an ordered tuple of real components.

A unit vector is a vector whose norm is one.

The cross product $a\times b$ of two three-dimensional vectors is perpendicular to both, has magnitude $|a||b|\sin \theta$, and is oriented by the right-hand rule.