Tangent, Normal and Binormal Vectors — Question 2

Question 2 – Solution

Strategy The parameter changes angular position while $a$ changes only the circle’s scale.

Computation $\vec r'=a\left\langle-\sin t,\cos t,0\right\rangle$, so $$\mathbf T=\left\langle-\sin t,\cos t,0\right\rangle,\quad \mathbf N=\left\langle-\cos t,-\sin t,0\right\rangle,\quad \mathbf B=\left\langle 0,0,1\right\rangle.$$

Interpretation The principal normal points radially inward regardless of radius. The curve lies in the fixed horizontal plane $z=h$, so the right-handed binormal is its constant upward normal.