Question

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 1

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:

`\rho = \sqrt{x^(2)+y^(2)+z^(2)}`

`\phi = cos^(-1)(z)/(\rho)`

For angle θ:

• If x > 0 and y > 0:
  `\theta = tan^(-1)(y)/(x)`;

• If x < 0:
  `\theta = \pi + tan^(-1)(y)/(x)`;
• If x > 0 and y < 0:
  `\theta = 2\pi + tan^(-1)(y)/(x)`;

Calculating:

a) (4,2,-4)

`\rho = \sqrt{4^(2)+2^(2)+(-4)^(2)}` = 6

`\phi = cos^(-1)((-4)/(6))`

`\phi = cos^(-1)((-2)/(3))`

For θ, choose 1st option:

`\theta = tan^(-1)((2)/(4))`

`\theta = tan^(-1)((1)/(2))`

b) (0,8,15)

`\rho = \sqrt{0^(2)+8^(2)+(15)^(2)}` = 17

`\phi = cos^(-1)((15)/(17))`

`\theta = tan^(-1)(y)/(x)`

The angle θ gives a tangent that doesn't exist. Analysing table of sine, cosine and tangent: θ =
`(\pi)/(2)`

c) (√2,1,1)

`\rho = \sqrt{(√(2) )^(2)+1^(2)+1^(2)}` = 2

`\phi = cos^(-1)((1)/(2))`

`\phi` =
`(\pi)/(3)`

`\theta = tan^(-1)(1)/(√(2) )`

d) (−2√3,−2,3)

`\rho = \sqrt{(-2√(3) )^(2)+(-2)^(2)+3^(2)}` = 5

`\phi = cos^(-1)((3)/(5))`

Since x < 0, use 2nd option:

`\theta = \pi + tan^(-1)(1)/(√(3) )`

`\theta = \pi + (\pi)/(6)`

`\theta = (7\pi)/(6)`

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

`r=\sqrt{x^(2)+y^(2)}`

Angle θ is the same as spherical coordinate;

z = z

Calculating:

a) (4,2,-4)

`r=\sqrt{4^(2)+2^(2)}` =
`√(20)`

`\theta = tan^(-1)(1)/(2)`

z = -4

b) (0, 8, 15)

`r=\sqrt{0^(2)+8^(2)}` = 8

`\theta = (\pi)/(2)`

z = 15

c) (√2,1,1)

`r=\sqrt{(√(2) )^(2)+1^(2)}` =
`√(3)`

`\theta = (\pi)/(3)`

z = 1

d) (−2√3,−2,3)

`r=\sqrt{(-2√(3) )^(2)+(-2)^(2)}` = 4

`\theta = (7\pi)/(6)`

z = 3