Convert $\left( 1 , \frac { \pi } { 4 } , \frac { \pi } { 4 } \right)$ from spherical coordinates to rectangular coordinates.
A) $\left( \frac { 1 } { 2 } , \frac { 1 } { 2 } , \frac { 1 } { 2 } \right)$
B) $\left( \frac { 1 } { 2 } , \frac { 1 } { 2 } , \frac { 1 } { \sqrt { 2 } } \right)$
C) $\left( \frac { 1 } { 2 } , \frac { 1 } { \sqrt { 2 } } , \frac { 1 } { 2 } \right)$
D) $\left( \frac { 1 } { \sqrt { 2 } } , \frac { 1 } { 2 } , \frac { 1 } { 2 } \right)$
E) $\left( 0 , \frac { 1 } { \sqrt { 2 } } , \frac { 1 } { \sqrt { 2 } } \right)$
F) $\left( \frac { 1 } { \sqrt { 2 } } , 1 , \frac { 1 } { \sqrt { 2 } } \right)$
G) $\left( \frac { 1 } { \sqrt { 2 } } , \frac { 1 } { \sqrt { 2 } } , 0 \right)$
H) $( 1,0,0 )$

Question

Convert $\left( 1 , \frac { \pi } { 4 } , \frac { \pi } { 4 } \right)$ from spherical coordinates to rectangular coordinates.
A) $\left( \frac { 1 } { 2 } , \frac { 1 } { 2 } , \frac { 1 } { 2 } \right)$ 
B) $\left( \frac { 1 } { 2 } , \frac { 1 } { 2 } , \frac { 1 } { \sqrt { 2 } } \right)$ 
C) $\left( \frac { 1 } { 2 } , \frac { 1 } { \sqrt { 2 } } , \frac { 1 } { 2 } \right)$ 
D) $\left( \frac { 1 } { \sqrt { 2 } } , \frac { 1 } { 2 } , \frac { 1 } { 2 } \right)$ 
E) $\left( 0 , \frac { 1 } { \sqrt { 2 } } , \frac { 1 } { \sqrt { 2 } } \right)$ 
F) $\left( \frac { 1 } { \sqrt { 2 } } , 1 , \frac { 1 } { \sqrt { 2 } } \right)$ 
G) $\left( \frac { 1 } { \sqrt { 2 } } , \frac { 1 } { \sqrt { 2 } } , 0 \right)$ 
H) $( 1,0,0 )$

Quizplus · Accepted Answer

The conversion from spherical coordinates $(\rho, \theta, \phi)$ to rectangular coordinates $(x, y, z)$ is given by $x = \rho \sin(\phi) \cos(\theta)$, $y = \rho \sin(\phi) \sin(\theta)$, and $z = \rho \cos(\phi)$. Substituting $\rho = 1$, $\theta = \frac{\pi}{4}$, and $\phi = \frac{\pi}{4}$ into these equations gives $x = \frac{1}{2}$, $y = \frac{1}{2}$, and $z = \frac{1}{\sqrt{2}}$.

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Convert $\left( 1 , \frac { \pi } { 4 } , \frac { \pi } { 4 } \right)$