Find the rectangular coordinates of the point, whose cylindrical coordinates are given.
(a)
(4, π/4, 1)
(x, y, z) =
(b)
(6, −π/3, 5)
(x, y, z) =
To find the rectangular coordinates (x, y, z) of a point given its cylindrical coordinates (r, θ, z), you can use the following formulas:
x = r * cos(θ)
y = r * sin(θ)
z = z
Let's use these formulas to find the rectangular coordinates for the given cylindrical coordinates.
(a)
Using the formulas:
x = 4 * cos(π/4) = 4 * √2/2 = 2√2
y = 4 * sin(π/4) = 4 * √2/2 = 2√2
z = 1
Therefore, (x, y, z) = (2√2, 2√2, 1).
(b)
Using the formulas:
x = 6 * cos(-π/3) = 6 * 1/2 = 3
y = 6 * sin(-π/3) = 6 * -√3/2 = -3√3
z = 5
Therefore, (x, y, z) = (3, -3√3, 5).