[Math] Derivative of cross-product of two vectors

calculusmultivariable-calculusvector analysis

In finding the derivative of the cross product of two vectors $\frac{d}{dt}[\vec{u(t)}\times \vec{v(t)}]$, is it possible to find the cross-product of the two vectors first before differentiating?

Best Answer

You can evaluate this expression in two ways:

  • You can find the cross product first, and then differentiate it.
  • Or you can use the product rule, which works just fine with the cross product:

$$ \frac{d}{dt}(\mathbf{u} \times \mathbf{v}) = \frac{d\mathbf{u}}{dt} \times \mathbf{v} + \mathbf{u} \times \frac{d\mathbf{v}}{dt} $$

Picking a method depends on the problem at hand. For example, the product rule is used to derive Frenet Serret formulas.

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