Math Problem Statement

Convert the point from cylindrical coordinates to rectangular coordinates: (3, -3π/2, 6)

Solution

To convert a point from cylindrical coordinates (r,θ,z)(r, \theta, z) to rectangular coordinates (x,y,z)(x, y, z), we use the following formulas:

x=rcos(θ)x = r \cdot \cos(\theta) y=rsin(θ)y = r \cdot \sin(\theta) z=zz = z

For the given cylindrical coordinates: (r,θ,z)=(3,3π2,6)(r, \theta, z) = \left(3, -\frac{3\pi}{2}, 6\right)

Let's apply the formulas:

  1. Calculate xx: x=3cos(3π2)x = 3 \cdot \cos\left(-\frac{3\pi}{2}\right)

    Since cos(3π2)=0\cos\left(-\frac{3\pi}{2}\right) = 0, we have: x=30=0x = 3 \cdot 0 = 0

  2. Calculate yy: y=3sin(3π2)y = 3 \cdot \sin\left(-\frac{3\pi}{2}\right)

    Since sin(3π2)=1\sin\left(-\frac{3\pi}{2}\right) = 1, we have: y=31=3y = 3 \cdot 1 = 3

  3. Calculate zz: z=6z = 6

So, the rectangular coordinates are: (x,y,z)=(0,3,6)(x, y, z) = (0, 3, 6)

Would you like further details or have any questions?

Related Questions

  1. How do cylindrical and spherical coordinates relate?
  2. What are common applications of cylindrical coordinates in physics?
  3. How does converting to polar coordinates differ in 2D versus 3D?
  4. How do I convert rectangular coordinates back to cylindrical coordinates?
  5. What are the advantages of using cylindrical coordinates in certain problems?

Tip: Always remember that cylindrical coordinates are often used when there's rotational symmetry about an axis, like in cases involving circular shapes or pipes.

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Math Problem Analysis

Mathematical Concepts

Coordinate Conversion
Cylindrical Coordinates
Rectangular Coordinates

Formulas

x = r * cos(θ)
y = r * sin(θ)
z = z

Theorems

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Suitable Grade Level

College or High School