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 express the given point in terms of radius (r), azimuthal angle (θ), and height (z).
For point (a) (3, -3, 2), we can follow these steps:
1. Calculate r: The distance from the origin to the given point is the radius r. Using the Pythagorean theorem, r = sqrt(x^2 + y^2), where x and y are the coordinates in the x-y plane. In this case, r = sqrt(3^2 + (-3)^2) = sqrt(18).
2. Calculate θ: The angle θ represents the azimuthal angle in the x-y plane. To find θ, we can use the formula θ = arctan(y / x). In this case, θ = arctan((-3) / 3) = arctan(-1) = -π/4 (since θ must be between 0 and 2π, we add 2π to get the equivalent positive angle θ = 7π/4).
3. z remains the same, so in this case, z = 2.
Therefore, the cylindrical coordinates for point (a) are (r, θ, z) = (sqrt(18), 7π/4, 2).
For point (b) (-2, -2√3, 5), the steps are similar:
1. r = sqrt((-2)^2 + (-2√3)^2) = sqrt(4 + 12) = sqrt(16) = 4.
2. θ = arctan((-2√3) / -2) = arctan(√3) = π/3.
3. z = 5.
Thus, the cylindrical coordinates for point (b) are (r, θ, z) = (4, π/3, 5).