Spherical Coordinates — Question 3

Question 3 – Solution

Strategy. Interpret $\rho$ as distance, $\theta$ as azimuth, and $\phi$ as inclination from the positive $z$-axis.

Step 1: Surfaces $\rho=5$ is the sphere of radius $5$. The equation $\theta=2\pi/3$ is a vertical half-plane from the $z$-axis in azimuth $2\pi/3$. The equation $\phi=\pi/4$ is the upper nappe of the cone making angle $\pi/4$ with the positive $z$-axis.

Step 2: Pairwise intersections Sphere–half-plane and sphere–cone intersections are semicircle and horizontal circle, respectively. Half-plane–cone is a ray from the origin.

Step 3: Common point Substitution gives $$x=5(\tfrac{\sqrt 2}{2})(-\tfrac 12)=-\frac{5\sqrt 2}{4},\quad y=5(\tfrac{\sqrt 2}{2})(\tfrac{\sqrt 3}{2})=\frac{5\sqrt 6}{4},\quad z=\frac{5\sqrt 2}{2}.$$ Thus $$\boxed{P=(-5\sqrt 2/4,\ 5\sqrt 6/4,\ 5\sqrt 2/2)}.$$