Change each of the following points from rectangular coordinates to spherical coordinates and to cylindrical coordinates. a. (4,2,−4)
b. (0,8,15)
c. (√2,1,1)
d. (−2√3,−2,3)

Answer and Step-by-step explanation: Spherical coordinate describes a location of a point in space: one distance (ρ) and two angles (Ф,θ).To transform cartesian coordinates into spherical coordinates:

For angle θ:

Calculating:

a) (4,2,-4)

= 6

For θ, choose 1st option:

b) (0,8,15)

= 17

The angle θ gives a tangent that doesn't exist. Analysing table of sine, cosine and tangent: θ =

c) (√2,1,1)

= 2

=

d) (−2√3,−2,3)

= 5

Since x < 0, use 2nd option:

Cilindrical coordinate describes a 3 dimension space: 2 distances (r and z) and 1 angle (θ). To express cartesian coordinates into cilindrical:

Angle θ is the same as spherical coordinate;

z = z

Calculating:

a) (4,2,-4)

=

z = -4

b) (0, 8, 15)

= 8

z = 15

c) (√2,1,1)

=

z = 1

d) (−2√3,−2,3)

= 4

z = 3