Tangential and normal components of acceleration

In the special case when n=3 we have cross products available and can find the tangential and normal components of acceleration easily:

$$\v{p''}(t) = \frac{\v{p}''(t) \cdot \v{p}'(t)}{\vert\v{p'}(t)... ...{p}''(t) \times \v{p}'(t)\vert}{\vert\v{p'}(t)\vert} \v{\bf N}$$

This makes it easy to use vector methods to find the second derivative of arc length and the curvature:

$$\begin{eqnarray*}\frac{d^2s}{dt^2} &=& \frac{\v{p''}(t) \cdot \v{p'}(t)}{\vert\v... ...rac{\vert\v{p''}(t) \times \v{p'}(t)\vert}{\vert\v{p'}(t)\vert^3}\end{eqnarray*}$$