Tangent, Normal and Binormal Vectors — Question 3

Question 3

Find the Frenet frame for $\vec r(t)=\left\langle t,t^2,t^3\right\rangle$ at $t=1$. Use the cross-product method for $\mathbf B$ and verify the orientation.

Question 3 – Solution

Strategy Use $\mathbf T=\vec r'/\|\vec r'\|$, $\mathbf B=(\vec r'\times\vec r'')/\|\vec r'\times\vec r''\|$, and $\mathbf N=\mathbf B\times\mathbf T$.

Derivatives At $t=1$, $\vec r'=\left\langle 1,2,3\right\rangle$ and $\vec r''=\left\langle 0,2,6\right\rangle$. Thus $$\mathbf T=\frac 1{\sqrt{14}}\left\langle 1,2,3\right\rangle,\qquad \mathbf B=\frac 1{\sqrt{19}}\left\langle 3,-3,1\right\rangle.$$

Normal Therefore $$\boxed{\mathbf N=\mathbf B\times\mathbf T=\frac 1{\sqrt{266}}\left\langle-11,-8,9\right\rangle}.$$

Verification Direct cross multiplication gives $\mathbf T\times\mathbf N=\mathbf B$ and all norms equal 1.