Convert $\left( 2 , \frac { 5 \pi } { 4 } , 3 \right)$ from cylindrical coordinates to rectangular coordinates.
A) $( 1,1,3 )$
B) $( 0,2,3 )$
C) $( 2,0,3 )$
D) $( \sqrt { 2 } , \sqrt { 2 } , 3 )$
E) $( - 1 , - 1,3 )$
F) $( 0 , - 2,3 )$
G) $( - 2,0,3 )$
H) $( - \sqrt { 2 } , - \sqrt { 2 } , 3 )$

Question

Convert $\left( 2 , \frac { 5 \pi } { 4 } , 3 \right)$ from cylindrical coordinates to rectangular coordinates.
A) $( 1,1,3 )$ 
B) $( 0,2,3 )$ 
C) $( 2,0,3 )$ 
D) $( \sqrt { 2 } , \sqrt { 2 } , 3 )$ 
E) $( - 1 , - 1,3 )$ 
F) $( 0 , - 2,3 )$ 
G) $( - 2,0,3 )$ 
H) $( - \sqrt { 2 } , - \sqrt { 2 } , 3 )$

Quizplus · Accepted Answer

The conversion from cylindrical to rectangular coordinates is given by $x = r\cos(\theta)$, $y = r\sin(\theta)$, and $z = z$. For $\left( 2 , \frac { 5 \pi } { 4 } , 3 \right)$, $r = 2$, $\theta = \frac { 5 \pi } { 4 }$, and $z = 3$. Thus, $x = 2\cos\left(\frac { 5 \pi } { 4 }\right) = -\sqrt{2}$ and $y = 2\sin\left(\frac { 5 \pi } { 4 }\right) = -\sqrt{2}$, leading to the coordinates $\left( - \sqrt { 2 } , - \sqrt { 2 } , 3 \right)$.

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