Vector product

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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) ISBN 1421407949, 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