Question

Convert the following Cartesian points to cylindrical and spherical coordinates: P(2, 5, 1)

Answer 1

To convert the Cartesian coordinates (x, y, z) to cylindrical coordinates (r, θ, z), you can use the formulas r = √(x² + y²), θ = arctan(y/x), and z = z. For the point P(2, 5, 1), the cylindrical coordinates would be r = √29, θ = 1.1908 radians, and z = 1. To convert the Cartesian coordinates to spherical coordinates (ρ, θ, φ), you can use the formulas ρ = √(x² + y² + z²), θ = arctan2(y, x), and φ = arccos(z/ρ). For the point P(2, 5, 1), the spherical coordinates would be ρ = √30, θ = 1.1908 radians, and φ = 1.2915 radians.

Step-by-step explanation:

To convert the Cartesian coordinates (x, y, z) to cylindrical coordinates (r, θ, z), we use the following formulas:

1. r = √(x² + y²)
2. θ = arctan(y/x)
3. z = z

Therefore, for the point P(2, 5, 1), we have:

1. r = √(2² + 5²) = √29
2. θ = arctan(5/2) = 1.1908 radians
3. z = 1

To convert the Cartesian coordinates (x, y, z) to spherical coordinates (ρ, θ, φ), we use the following formulas:

1. ρ = √(x² + y² + z²)
2. θ = arctan2(y, x)
3. φ = arccos(z/ρ)

Therefore, for the point P(2, 5, 1), we have:

1. ρ = √(2² + 5² + 1²) = √30
2. θ = arctan2(5, 2) = 1.1908 radians
3. φ = arccos(1/√30) = 1.2915 radians