Question

Obtain the spherical coordinates of the following points, specified in terms of their Cartesian coordinates.

a. (x,y,z)=(1,0,0)
b. (x,y,z)=(0,2,0)
c. (x,y,z)=(0,0,5)
d. (x,y,z)=(3,3,0)
e. (x,y,z)=(0,−3,4)

Answer 1

To obtain the spherical coordinates of points given in Cartesian coordinates, we use formulas involving r, θ, and φ. Applying these formulas to each given point, we find their respective spherical coordinates.

Step-by-step explanation:

To obtain the spherical coordinates of a point given in Cartesian coordinates, we use the following formulas:

r = sqrt(x^2 + y^2 + z^2)

θ = tan^(-1)(sqrt(x^2 + y^2) / z)

φ = tan^(-1)(y / x)

Using these formulas, we can calculate the spherical coordinates for each given point:

a. (1,0,0): r = sqrt(1^2 + 0^2 + 0^2) = 1, θ = tan^(-1)(sqrt(1^2 + 0^2) / 0) = tan^(-1)(0/0) (undefined), φ = tan^(-1)(0 / 1) = 0

b. (0,2,0): r = sqrt(0^2 + 2^2 + 0^2) = 2, θ = tan^(-1)(sqrt(0^2 + 2^2) / 0) = tan^(-1)(2/0) (undefined), φ = tan^(-1)(2 / 0) = tan^(-1)(∞) (undefined)

c. (0,0,5): r = sqrt(0^2 + 0^2 + 5^2) = 5, θ = tan^(-1)(sqrt(0^2 + 0^2) / 5) = tan^(-1)(0/5) = 0, φ = tan^(-1)(0 / 0) (undefined)

d. (3,3,0): r = sqrt(3^2 + 3^2 + 0^2) = sqrt(18) ≈ 4.2426, θ = tan^(-1)(sqrt(3^2 + 3^2) / 0) = tan^(-1)(6/0) (undefined), φ = tan^(-1)(3 / 3) = tan^(-1)(1) ≈ 0.7854

e. (0,-3,4): r = sqrt(0^2 + (-3)^2 + 4^2) = sqrt(25) = 5, θ = tan^(-1)(sqrt(0^2 + (-3)^2) / 4) = tan^(-1)(3/4) ≈ 0.6435, φ = tan^(-1)(-3 / 0) = tan^(-1)(-∞) ≈ -1.5708