Understanding step in deriving the formula of the curvature.

Question

Last formula on third page of the document:

• Computation of $\vec{r'}(t)\times \vec{r''}(t)$ From the previous two formulas and using the properties of cross products we see that

$$\vec{r'}\times \vec{r''}=\frac{ds}{dt}\frac{d^2s}{dt^2}\left(\vec{T} \times\vec{T}\right)+\left(\frac{ds}{dt}\right)^2\left(\vec{T}\times\vec{T}'\right)$$

What properties of cross products are being used here?

Thanks in advance.

Answer 1

All they're doing is distributing the product over the sum. Ie using:

\begin{equation} \vec{a} \times (\vec{b} + \vec{c}) = (\vec{a}\times\vec{b}) + (\vec{a}\times\vec{c}) \end{equation}

And then pulling all the scalars to the front of each term. Its pretty easy to see if you substitute in the "Computation of..." bits into $\vec{r}'\times\vec{r}''$.