Question
Convert the point \( (\rho, \theta, \phi)=\left(6, \frac{3 \pi}{4}, \pi\right) \) to Cartesian coordinates. Give answers either as expressions, or decimals to at least one decimal place. \( (x, y, z)=\square \) Question Help: \( \square \) Video 1
Ask by Summers Weaver. in the United States
Mar 08,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( (x, y, z) = (0, 0, -6) \)
Solution
The conversion formulas from spherical to Cartesian coordinates are
\[
x = \rho \sin(\phi) \cos(\theta), \quad y = \rho \sin(\phi) \sin(\theta), \quad z = \rho \cos(\phi).
\]
Given \(\rho = 6\), \(\theta = \frac{3\pi}{4}\), and \(\phi = \pi\):
1. Calculate \(x\):
\[
x = 6 \sin(\pi) \cos\left(\frac{3\pi}{4}\right).
\]
Since \(\sin(\pi) = 0\),
\[
x = 6 \cdot 0 \cdot \cos\left(\frac{3\pi}{4}\right) = 0.
\]
2. Calculate \(y\):
\[
y = 6 \sin(\pi) \sin\left(\frac{3\pi}{4}\right).
\]
Again, \(\sin(\pi) = 0\),
\[
y = 6 \cdot 0 \cdot \sin\left(\frac{3\pi}{4}\right) = 0.
\]
3. Calculate \(z\):
\[
z = 6 \cos(\pi).
\]
Since \(\cos(\pi) = -1\),
\[
z = 6 \cdot (-1) = -6.
\]
Thus, the Cartesian coordinates are
\[
(x, y, z) = (0, 0, -6).
\]
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Simplify this solution Mind Expander
To convert the spherical coordinates \( (\rho, \theta, \phi) = \left(6, \frac{3 \pi}{4}, \pi\right) \) to Cartesian coordinates, you can use the following formulas: \[ x = \rho \sin(\phi) \cos(\theta) \] \[ y = \rho \sin(\phi) \sin(\theta) \] \[ z = \rho \cos(\phi) \] Plugging in the values, we find: - For \( z \), \( z = 6 \cos(\pi) = 6 \times (-1) = -6 \). - For \( x \), \( x = 6 \sin(\pi) \cos\left(\frac{3 \pi}{4}\right) = 6 \times 0 \times (-\frac{\sqrt{2}}{2}) = 0 \). - For \( y \), \( y = 6 \sin(\pi) \sin\left(\frac{3 \pi}{4}\right) = 6 \times 0 \times \frac{\sqrt{2}}{2} = 0 \). Thus, the Cartesian coordinates are \( (x, y, z) = (0, 0, -6) \).
