Tangent, Normal and Binormal Vectors — Question 1

Question 1

For the helix $\vec r(t)=\left\langle\cos t,\sin t,t\right\rangle$, find the unit tangent $\mathbf T$, principal unit normal $\mathbf N$, and binormal $\mathbf B$ for all $t$. Evaluate the frame at $t=0$.

Question 1 – Solution

Strategy Normalize $\vec r'$, normalize $\mathbf T'$, then use $\mathbf B=\mathbf T\times\mathbf N$.

Frame Since $\|\vec r'\|=\sqrt 2$, $$\mathbf T=\frac 1{\sqrt 2}\left\langle-\sin t,\cos t,1\right\rangle,\qquad \mathbf N=\left\langle-\cos t,-\sin t,0\right\rangle,$$ and $$\mathbf B=\frac 1{\sqrt 2}\left\langle\sin t,-\cos t,1\right\rangle.$$

At $t=0$ $\boxed{\mathbf T=\frac 1{\sqrt 2}\left\langle 0,1,1\right\rangle,\ \mathbf N=\left\langle-1,0,0\right\rangle,\ \mathbf B=\frac 1{\sqrt 2}\left\langle 0,-1,1\right\rangle}$.

Verification The three vectors are unit, mutually orthogonal, and $\mathbf T\times\mathbf N=\mathbf B$.