Tangent, Normal and Binormal Vectors — Question 6

Question 6 – Solution

Strategy Use the frame from the helix and choose the Frenet vector perpendicular to each named plane.

Frame and point At $t=0$, $P=(1,0,0)$, $\mathbf T\parallel\left\langle 0,1,1\right\rangle$, $\mathbf N=\left\langle-1,0,0\right\rangle$, and $\mathbf B\parallel\left\langle 0,-1,1\right\rangle$.

Planes The normal plane is perpendicular to $\mathbf T$: $\boxed{y+z=0}$, and is spanned by $\mathbf N,\mathbf B$. The osculating plane is perpendicular to $\mathbf B$: $\boxed{-y+z=0}$, spanned by $\mathbf T,\mathbf N$. The rectifying plane is perpendicular to $\mathbf N$: $\boxed{x=1}$, spanned by $\mathbf T,\mathbf B$.