Tangent, Normal and Binormal Vectors — Question 1

Question 1 – Solution

Strategy. At a regular point, the unit tangent is $\mathbf T=\mathbf r'/\|\mathbf r'\|$.

Step 1: Point and velocity $$\mathbf r(1)=\left\langle 1,2,-1\right\rangle,\qquad \mathbf r'(t)=\left\langle 2t,2,2t-2\right\rangle,$$ so $\mathbf r'(1)=\left\langle 2,2,0\right\rangle\ne\mathbf 0$.

Step 2: Normalize $$\|\mathbf r'(1)\|=\sqrt{2^2+2^2}=2\sqrt 2.$$ Therefore $$\boxed{\mathbf T(1)=\frac{\left\langle 2,2,0\right\rangle}{2\sqrt 2} =\left\langle \tfrac 1{\sqrt 2},\tfrac 1{\sqrt 2},0\right\rangle}.$$

Step 3: Tangent line Either $\mathbf r'(1)$ or the parallel unit vector may be used: $$\boxed{\mathbf L(s)=\left\langle 1,2,-1\right\rangle+s\left\langle 1,1,0\right\rangle}.$$ At $s=0$ the line passes through the curve point, and its direction $\left\langle 1,1,0\right\rangle$ is one-half of $\mathbf r'(1)$.

Verification. The squared length of $\mathbf T(1)$ is $1/2+1/2=1$, as required.