3.I.3C

A curve is given in terms of a parameter $t$ by

$\mathbf{x}(t)=\left(t-\frac{1}{3} t^{3}, t^{2}, t+\frac{1}{3} t^{3}\right)$

(i) Find the arc length of the curve between the points with $t=0$ and $t=1$.

(ii) Find the unit tangent vector at the point with parameter $t$, and show that the principal normal is orthogonal to the $z$ direction at each point on the curve.

*Typos? Please submit corrections to this page on GitHub.*