Vector product

From Encyclopedia of Mathematics
Jump to: navigation, search

of a vector $a$ by a vector $b$ in $\mathbb{R}^3$

The vector $c$, denoted by the symbol $a\times b$ or $[a,b]$, satisfying the following requirements:

  1. the length of $c$ is equal to the product of the lengths of the vectors $a$ and $b$ by the sine of the angle $\phi$ between them, i.e. \begin{equation} |c| = |a\times b| = |a|\cdot |b| \sin\phi; \end{equation}
  2. $c$ is orthogonal to both $a$ and $b$;
  3. the orientation of the vector triple $a,b,c$ is the same as that of the (standard) triple of basis vectors. See Vector algebra.


Let $a=(a_1,a_2,a_3)$ and $b=(b_1,b_2,b_3)$ have coordinates with respect to an orthonormal basis in $\mathbb{R}^3$, then the coordinates of $c=a\times b$ are \begin{equation}c=\begin{pmatrix} a_2 b_3 - a_3 b_2 \\ a_3 b_1 - a_1 b_3 \\ a_1 b_2 - a_2 b_1\end{pmatrix}.\end{equation}

The vector product is sometimes called cross product [1], also cf. cross product.


  1. Gene H. Golub, Charles F. Van Loan, Matrix Computations, Johns Hopkins Studies in the Mathematical Sciences 3, JHU Press (2013) Template:ISBN, p. 70.
How to Cite This Entry:
Vector product. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article