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Question

# Plot the point whose spherical coordinates are given. Then find the rectangular coordinates of the point. (3, π/2, 3π/4)

Solution

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Step 1
1 of 3

Given that

$\left(\rho, \theta, \phi\right) = \left(3, \dfrac{\pi}{2}, \dfrac{3\pi}{4} \right)$

Therefore

${\color{#c34632}x = \rho\sin\phi\cos\theta }= 3\sin\left(\dfrac{3\pi}{4} \right)\cos\left(\dfrac{\pi}{2} \right) = 0$

${\color{#c34632}y = \rho\sin\phi\sin\theta }= 3\sin\left(\dfrac{3\pi}{4} \right)\sin\left(\dfrac{\pi}{2} \right) = 3\left[ \dfrac{1}{\sqrt{2}}\right]\left[1 \right] =\dfrac{3\sqrt{2}}{2}$

${\color{#c34632}z = \rho\cos\phi }= 3\cos\left(\dfrac{3\pi}{4} \right) = 3\left[-\dfrac{1}{\sqrt{2}} \right] =-\dfrac{3\sqrt{2}}{2}$

The point is $\left(x, y, z \right)=\left( 0, \dfrac{3\sqrt{2}}{2}, -\dfrac{3\sqrt{2}}{2}\right)$

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