Calculus with Vector Functions — Question 4

Question 4

The curves $\vec r_1(t)=\left\langle t,t^2,t^3\right\rangle$ and $\vec r_2(s)=\left\langle s^2,s,s^3\right\rangle$ meet at the origin and at $(1,1,1)$. Find the smaller angle between their tangent lines at each intersection.

Question 4 – Solution

Strategy Differentiate each parametrization and evaluate using its own parameter at the shared point.

At the origin Both use parameter 0. The tangent vectors are $\vec r_1'(0)=\left\langle 1,0,0\right\rangle$ and $\vec r_2'(0)=\left\langle 0,1,0\right\rangle$, so the angle is $\boxed{90^\circ}$.

At $(1,1,1)$ Both use parameter 1. The tangents are $\left\langle 1,2,3\right\rangle$ and $\left\langle 2,1,3\right\rangle$. Their dot product is 13 and both magnitudes are $\sqrt{14}$, so $$\boxed{\theta=\cos^{-1}(13/14)\approx 21.8^\circ}.$$