Exercise 15

In this activity, you verified for a specific line that the curvature is $0$. The generalization of this is that the unit speed parameterization of a line through $(x_0,y_0,z_0)$ with direction vector $\langle a,b,c\rangle$ can be given by $$\begin{aligned} \vr(s)& =\langle x(s),y(s),z(s) \rangle \\ &= \left\langle x_0+\frac{a}{\sqrt{a^2+b^2+c^2}}s , y_0+\frac{b}{\sqrt{a^2+b^2}}s ,z_0+ \frac{c}{\sqrt{a^2+b^2+c^2}}s\right\rangle \end{aligned}$$.

1. Compute the speed of this parameterization to verify that this has unit speed. This will also show that $\vr\, '(s)=\vv(s)=\vT(s)$ (which is not true in general).

2. Calculate $\vT(s)$, $\frac{d\vT}{ds}$, and $\vecmag{ \frac{d\vT}{ds} }$ for this unit speed parameterization of a line.

3. Write a few sentences to explain why your result for the calculation of the curvature of a line makes sense in general.