A point has rectangular coordinates $(2,-1,-2)$ and spherical coordinates $(\rho, \theta, \phi).$  Find the rectangular coordinates of the point with spherical coordinates $(\rho, \theta, 2 \phi).$
Question:
A point has rectangular coordinates $(2,-1,-2)$ and spherical coordinates $(\rho, \theta, \phi).$  Find the rectangular coordinates of the point with spherical coordinates $(\rho, \theta, 2 \phi).$
Answer:
Let's think step by step.

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\left( -\frac{8}{3}, \frac{4}{3}, -\frac{1}{3} \right)
A point has rectangular coordinates $(2,-1,-2)$ and spherical coordinates $(\rho, \theta, \phi).$  Find the rectangular coordinates of the point with spherical coordinates $(\rho, \theta, 2 \phi).$
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