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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:

# 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}
# $c$ is orthogonal to both $a$ and $b$;
# the orientation of the vector triple $a,b,c$ is the same as that of the (standard) triple of basis vectors. See [[Vector algebra]].



====Comments====

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 <ref name="Matrix Computations" />, also cf. [[cross product]].



====References====

<references>
<ref name="Matrix Computations">Gene H. Golub, Charles F. Van Loan, ''Matrix Computations'', Johns Hopkins Studies in the Mathematical Sciences '''3''', JHU Press (2013) {{ISBN|1421407949}}, p. 70.</ref>
</references>