What are the cylindrical coordinates of the point whose rectangular coordinates are (x=−4, y=4, z=3)?
Cylindrical coordinates are used to represent curved or spherical surfaces on the graph. Representing them on the cartesian plane is very difficult since they are circular in nature. These coordinates are extensively used in multiple fields of engineering and science. These coordinates include three parameters: (r, θ, z).
Answer: The cylindrical coordinates of the point whose rectangular coordinates are (x = −4, y = 4, z = 3), are (4√2, 3π / 4, 3).
Let's understand how did we arrive at the solution.
To convert from cartesian coordinates (x, y, z) to cylindrical coordinates (r, θ, z), we use:
x = r cos θ, y = r sin θ
now if we square and add both x and y terms, we get:
x2 + y2 = r2cos2 θ + r2sin2 θ
= r2(cos2θ + sin2θ) = r2 (using trigonometric identity cos2 θ + sin2 θ = 1)
Hence, r = √(x2 + y2)
Here: x = -4, y = 4, z = 3
Using the above equations;
⇒r = √(42 + (-4)2) = 4√2
Also, using r cos θ = x, we get:
⇒cos θ = -1 / √2
⇒θ = 3π/4
We keep the z coordinate as it is.