Spherical Coordinates — Question 9

Question 9

A point $P$ has spherical coordinates $(\rho,\theta,\phi)$ with $\rho=10$ and is equally distant from the positive $x$-, $y$-, and $z$-axes. Assume $P$ lies in the first octant.

Tasks

1. Determine the Cartesian coordinates of $P$.

2. Find its principal spherical angles.

3. Verify the three axis distances directly.

Question 9 – Solution

Strategy. Express squared distance to each coordinate axis and use first-octant signs.

Step 1: Equal-distance conditions Squared distances to the $x$-, $y$-, and $z$-axes are $y^2+z^2$, $x^2+z^2$, and $x^2+y^2$. Equality implies $x^2=y^2=z^2$. In the first octant, $x=y=z$.

Step 2: Radius condition Since $x^2+y^2+z^2=100$, we obtain $3x^2=100$ and $$\boxed{P=(10/\sqrt 3,10/\sqrt 3,10/\sqrt 3)}.$$ Then $\theta=\pi/4$, and $\cos\phi=z/\rho=1/\sqrt 3$, so $$\boxed{(\rho,\theta,\phi)=(10,\pi/4,\arccos(1/\sqrt 3))}.$$

Verification. Every squared axis distance is $100/3+100/3=200/3$, so every distance is $\boxed{10\sqrt{2/3}}$.