Curvature — Question 1

Question 1 – Solution

Strategy. The cross-product formula avoids differentiating the unit tangent.

Step 1: Derivatives $$\mathbf r'(t)=\left\langle 1,2t,0\right\rangle,\qquad \mathbf r''(t)=\left\langle 0,2,0\right\rangle.$$ At $t=1$, $\mathbf r'(1)=\left\langle 1,2,0\right\rangle$ and $\|\mathbf r'(1)\|=\sqrt 5$.

Step 2: Cross product $$\mathbf r'(1)\times\mathbf r''(1) =\left\langle 1,2,0\right\rangle\times\left\langle 0,2,0\right\rangle=\left\langle 0,0,2\right\rangle,$$ so its magnitude is $2$. Therefore $$\boxed{\kappa(1)=\frac 2{(\sqrt 5)^3}=\frac{2}{5\sqrt 5}}.$$

Step 3: Radius of curvature Since $\rho=1/\kappa$, $$\boxed{\rho(1)=\frac{5\sqrt 5}{2}}.$$ Both values are positive, and their product is $1$, verifying the reciprocal relationship.