Difference between revisions of "Vector product"
From Encyclopedia of Mathematics
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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]]. | ||
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| − | 3 | + | |
| + | ====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> | ||
Latest revision as of 05:53, 15 April 2023
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 [1], also cf. cross product.
References
- ↑ 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: http://encyclopediaofmath.org/index.php?title=Vector_product&oldid=13392
Vector product. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Vector_product&oldid=13392
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article