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Cross Product

For Vectors ${\bf u}$ and ${\bf v}$,

{\bf u}\times{\bf v} = \hat {\bf x} (u_yv_z-u_zv_y)-\hat {\bf y}(u_xv_z-u_zv_x)+ \hat{\bf z}(u_xv_y-u_yv_x).
\end{displaymath} (1)

This can be written in a shorthand Notation which takes the form of a Determinant
{\bf u}\times{\bf v} = \left\vert\matrix{
\hat {\bf x} & \h...
...\bf z}\cr
u_x & u_y & u_z\cr
v_x & v_y & v_z\cr}\right\vert.
\end{displaymath} (2)

It is also true that
$\displaystyle \vert{\bf u}\times{\bf v}\vert$ $\textstyle =$ $\displaystyle \vert{\bf u}\vert\, \vert{\bf v}\vert\sin\theta,$ (3)
  $\textstyle =$ $\displaystyle \vert{\bf u}\vert\, \vert{\bf v}\vert\sqrt{1-(\hat{\bf u}\cdot\hat{\bf v})^2},$ (4)

where $\theta$ is the angle between ${\bf u}$ and ${\bf v}$, given by the Dot Product
\cos\theta\equiv \hat{\bf u}\cdot\hat{\bf v}.
\end{displaymath} (5)

Identities involving the cross product include

{d\over dt} [{\bf r}_1(t)\times {\bf r}_2(t)]
= {\bf r}_1(t...
...d{\bf r}_2\over dt} + {d{\bf r}_1\over dt} \times {\bf r}_2(t)
\end{displaymath} (6)

{\bf A}\times {\bf B} = -{\bf B}\times {\bf A}
\end{displaymath} (7)

{\bf A}\times ({\bf B}+{\bf C}) = {\bf A}\times {\bf B}+{\bf A}\times {\bf C}
\end{displaymath} (8)

(t{\bf A})\times {\bf B} = t({\bf A}\times {\bf B}).
\end{displaymath} (9)

For a proof that ${\bf A}\times{\bf B}$ is a Pseudovector, see Arfken (1985, pp. 22-23). In Tensor notation,
{\bf A}\times{\bf B} = \epsilon_{ijk}A^jB^k,
\end{displaymath} (10)

where $\epsilon_{ijk}$ is the Levi-Civita Tensor.

See also Dot Product, Scalar Triple Product


Arfken, G. ``Vector or Cross Product.'' §1.4 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 18-26, 1985.

© 1996-9 Eric W. Weisstein