Question

Find The Rectangular Coordinates Of The Point, Whose Cylindrical Coordinates Are Given. (A) (2,Π,E) (B) (1,3π/2,3)Use Spherical Coordinates.

Answer 1

Let's solve this step-by-step:

(A) We're given (r, θ, z) = (2, Π, E) in cylindrical coordinates.

Step 1: We start off with (r, θ, z) = (2, Π, E)

Step 2: Conversion from cylindrical to rectangular coordinates involves using the following formulas
x = r cos(θ)
y = r sin(θ)

Step 3: Plug and calculate
x = r cos(θ) = 2 * cos(Π) = -2.0
y = r sin(θ) = 2 * sin(Π) = 2.4492935982947064e-16 (which is virtually zero)
The z-coordinate remains the same in both systems so z = E = 2.718281828459045

So the point in rectangular coordinates is (-2.0, 2.4492935982947064e-16, 2.718281828459045)

(B) We're given (r, θ, z) = (1, 3π/2, 3) in cylindrical coordinates.

Step 1: We start off with (r, θ, z) = (1, 3π/2, 3)

Step 2: Conversion from cylindrical to rectangular coordinates involves using the following formulas
x = r cos(θ)
y = r sin(θ)

Step 3: Plug and calculate
x = r cos(θ) = 1 * cos(3Π/2) = -1.8369701987210297e-16 (which is virtually zero)
y = r sin(θ) = 1 * sin(3Π/2) = -1
The z-coordinate remains the same in both systems so z = 3

So the point in rectangular coordinates is (-1.8369701987210297e-16, -1, 3)

And there's your answer. The rectangular coordinates for point A are (-2.0, 2.4492935982947064e-16, 2.718281828459045) and for point B are (-1.8369701987210297e-16, -1, 3).