Deriving Cartesian Coordinates from Cylindrical Coordinates

Question

The ans given was:

• $x = r \cos (\alpha)$
• $y = h$
• $z = -r \sin(\alpha)$

Could somebody explain to me how to arrive at the formula?

I'm probably confused with the axes because usually, the $Z$ axis was given as the depth or "height" for cylindrical coordinates. Even when the axes rotates, shouldn't the formula be

• $x = r \cos(\alpha)$
• $y = r \sin(\alpha)$
• $z = z$

Sorry if this qns seems elementary. I'm really confused with 3D coordinates

Answer 1

The important facts of cylindrical coordinates are:

1. You need an axis to determine height,
2. The other two coordinates will be given as distance from the origin and rotation around the other planes.

Therefore, height ("depth") doesn't have to be $z$. Here is how I would go about it:

The picture suggests you use the $y$ axis as height, therefore $y=h$. We need to a point in the $xz$ plane in polar coordinates. From the picture you can see that the vector has a positive $x$ component and a negative $z$ component, hence we perform $x = r \cos (\alpha)$ and $z = - r \sin (\alpha)$.