Change from rectangular to cylindrical coordinates. (Let r ≥ 0 and 0 ≤ θ ≤ 2π.)
(a)
(3, −3, 2)
(b)
(−2, −2sqrt3, 5)
To change from rectangular to cylindrical coordinates, we need to use the following formulas:
r = √(x^2 + y^2)
θ = arctan(y/x)
z = z
Let's calculate the cylindrical coordinates for each point:
(a) (3, -3, 2)
r = √(3^2 + (-3)^2) = √(9 + 9) = √18 = 3√2
θ = arctan((-3)/3) = arctan(-1) = -π/4 (since θ is between 0 and 2π, we note that -π/4 is equivalent to 7π/4)
z = 2
So in cylindrical coordinates, (3, -3, 2) is represented as (3√2, 7π/4, 2).
(b) (-2, -2√3, 5)
r = √((-2)^2 + (-2√3)^2) = √(4 + 12) = √16 = 4
θ = arctan((-2√3)/(-2)) = arctan(√3) = π/3 (since θ is between 0 and 2π)
z = 5
So in cylindrical coordinates, (-2, -2√3, 5) is represented as (4, π/3, 5).
To change from rectangular coordinates to cylindrical coordinates, we need to determine the radius (r), the angle (θ), and the height (z) in the cylindrical coordinate system.
(a)
To convert (3, -3, 2) from rectangular coordinates to cylindrical coordinates:
- The radius (r) can be found using the formula: r = √(x^2 + y^2), where x and y are the rectangular coordinates. In this case, r = √(3^2 + (-3)^2) = √(9 + 9) = √18 = 3√2.
- The angle (θ) can be found using the formula: θ = arctan(y/x), where x and y are the rectangular coordinates. In this case, θ = arctan((-3)/3) = arctan(-1) = -π/4.
- The height (z) remains the same, which is z = 2.
Therefore, the cylindrical coordinates for (3, -3, 2) are (3√2, -π/4, 2).
(b)
To convert (-2, -2√3, 5) from rectangular coordinates to cylindrical coordinates:
- The radius (r) can be found using the formula: r = √(x^2 + y^2), where x and y are the rectangular coordinates. In this case, r = √((-2)^2 + (-2√3)^2) = √(4 + 12) = √16 = 4.
- The angle (θ) can be found using the formula: θ = arctan(y/x), where x and y are the rectangular coordinates. In this case, θ = arctan((-2√3)/(-2)) = arctan(√3) = π/3.
- The height (z) remains the same, which is z = 5.
Therefore, the cylindrical coordinates for (-2, -2√3, 5) are (4, π/3, 5).
same as polar coordinates:
r^2 = x^2+y^2
tanθ = y/x