Vector Calculus | Part IA, 2008

A curve is given in terms of a parameter tt by

x(t)=(t13t3,t2,t+13t3)\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=0t=0 and t=1t=1.

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

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