Spherical Coordinates — Question 7

Question 7 – Solution

Strategy. Take the dot product of the two unit radial directions and apply the three-dimensional law of cosines.

Step 1: Direction angle If $\gamma$ is the angle between the radial directions, then $$\cos\gamma=\cos\phi_1\cos\phi_2+\sin\phi_1\sin\phi_2\cos(\theta_1-\theta_2).$$ Here $\cos\phi_1\cos\phi_2=-1/4$, $\sin\phi_1\sin\phi_2=3/4$, and $\cos(-2\pi/3)=-1/2$, so $\cos\gamma=-5/8$.

Step 2: Distance Therefore $$d^2=\rho_1^2+\rho_2^2-2\rho_1\rho_2\cos\gamma=9+25-30(-5/8)=\frac{211}{4}.$$ Thus $$\boxed{d=\frac{\sqrt{211}}{2}}.$$

Verification. The triangle inequality requires $|5-3|\le d\le 5+3$. Numerically $d\approx 7.26$, which lies between $2$ and $8$.