Cross Product

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

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

This can be written in a shorthand Notation which takes the form of a Determinant

$${\bf u}\times{\bf v} = \left\vert\matrix{ \hat {\bf x} & \h... ...\bf z}\cr u_x & u_y & u_z\cr v_x & v_y & v_z\cr}\right\vert.$$ (2)

It is also true that

$$\vert{\bf u}\times{\bf v}\vert = \vert{\bf u}\vert\, \vert{\bf v}\vert\sin\theta,$$ (3)

$$= \vert{\bf u}\vert\, \vert{\bf v}\vert\sqrt{1-(\hat{\bf u}\cdot\hat{\bf v})^2},$$ (4)

where $\theta$ is the angle between ${\bf u}$ and ${\bf v}$, given by the Dot Product

$$\cos\theta\equiv \hat{\bf u}\cdot\hat{\bf v}.$$ (5)

Identities involving the cross product include

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

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

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

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

For a proof that ${\bf A}\times{\bf B}$ is a Pseudovector, see Arfken (1985, pp. 22-23). In Tensor notation,

$${\bf A}\times{\bf B} = \epsilon_{ijk}A^jB^k,$$ (10)

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

See also Dot Product, Scalar Triple Product

References

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