Derivative of Vector Cross Product of Vector-Valued Functions

Theorem
Let:


 * $\mathbf r: t \mapsto \begin {bmatrix} x \\ y \\ z \end{bmatrix}$


 * $\mathbf q: t \mapsto \begin {bmatrix} \chi \\ \gamma \\ \zeta \end{bmatrix}$

be differentiable vector-valued functions, where:


 * $x, y, z, \chi, \gamma, \zeta$

are (images of) differentiable real functions.

The derivative of the vector cross product of $\mathbf r$ and $\mathbf q$ is given by:


 * $D_t \left({\mathbf r \left({t}\right) \times \mathbf q \left({t}\right)}\right) = \mathbf r' \left({x}\right) \times \mathbf q \left({x}\right) + \mathbf r \left({x}\right) \times \mathbf q'\left({x}\right)$

Also see

 * Derivative of Dot Product of Vector-Valued Functions
 * Derivative of Product of Real Function and Vector-Valued Function