Question
Convert the point \( (r, \theta, z)=\left(6, \frac{\pi}{2}, 4\right) \) to Cartesian coordinates. Give answers either as expressions, or decimals to at least one decimal place.
Ask by Simmons Conner. in the United States
Jan 27,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The Cartesian coordinates are (0, 6, 4).
Solution
To convert the cylindrical coordinates \((r, \theta, z) = \left(6, \frac{\pi}{2}, 4\right)\) to Cartesian coordinates \((x, y, z)\), use the following formulas:
\[
\begin{align*}
x &= r \cos \theta \\
y &= r \sin \theta \\
z &= z
\end{align*}
\]
Substituting the given values:
\[
\begin{align*}
x &= 6 \cos\left(\frac{\pi}{2}\right) = 6 \times 0 = 0 \\
y &= 6 \sin\left(\frac{\pi}{2}\right) = 6 \times 1 = 6 \\
z &= 4
\end{align*}
\]
Therefore, the Cartesian coordinates are:
\[
(x, y, z) = (0, 6, 4)
\]
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Simplify this solution Mind Expander
To convert the point \((r, \theta, z) = \left(6, \frac{\pi}{2}, 4\right)\) to Cartesian coordinates \((x, y, z)\), we can use the following relationships: 1. \(x = r \cdot \cos(\theta)\) 2. \(y = r \cdot \sin(\theta)\) 3. \(z = z\) Plugging in the given values: - \(r = 6\) - \(\theta = \frac{\pi}{2}\) - \(z = 4\) Calculating \(x\) and \(y\): \[ x = 6 \cdot \cos\left(\frac{\pi}{2}\right) = 6 \cdot 0 = 0 \] \[ y = 6 \cdot \sin\left(\frac{\pi}{2}\right) = 6 \cdot 1 = 6 \] So, the Cartesian coordinates are: \[ (x, y, z) = (0, 6, 4) \] Therefore, the point in Cartesian coordinates is \((0, 6, 4)\).
