Question

Convert point P(1, 3, 5) from Cartesian to cylindrical and spherical coordinate.

Answer 1

Concept:

Cartesian coordinates are a type of coordinate system that allows you to express the location of a point in the plane or in three dimensions. In three-dimensional space, cylindrical coordinates are an extension of polar coordinates. The polar coordinates in the x-y plane are combined with the z coordinate of cartesian coordinates.

Given:

The point P(1,3,5).

Find:

Convert Cartesian coordinates to cylindrical coordinates and spherical coordinates.

Solution:

The conversion of cartesian coordinates (x,y,z) to cylindrical coordinates (r,θ, z) are as follows,

r = √(x²+y²,

tanθ = y/x,

z = z.

The point P(1, 3, 5) to cylindrical coordinates is

$r = \sqrt{1^2 +3^2 }\\ r = \sqrt{1+9}\\ r = \sqrt{10}$

tanθ = 3/1,

tanθ = 3.

θ = 71.5 degrees.

z = 5.

The conversion of cartesian coordinates (x,y,z) to spherical coordinates (r,θ, Ф) are as follows,

r = √(x²+y²+z²,

tanθ = y/x,

Ф = $tan^{-1}(\frac{\sqrt{x^2+y^2}}{z})$.

The point P(1, 3, 5) to spherical coordinates is:

$r = \sqrt{x^2+y^2+z^2}\\ r = \sqrt{1^2+3^2+5^2}\\ r=\sqrt{1+9+25}\\ r = \sqrt{35}\\ r=5.91$

tanθ = 3/1,

tanθ = 3.

θ = 71.5 degrees.

Ф =$tan^{-1}(\frac{\sqrt{1^2+3^2}}{5})$

   = $tan^{-1}(\frac{\sqrt{10}}{5})$

   = 32.3 degrees.

Hence, the cylindrical coordinates are (√10, 71.5°, 5) and the spherical coordinates are (5.91,  71.5°, 32.3°).