This page contains a utility that calculates the 3D coordinates of a point in coordinate reference frames that are rotated in succession, first, around the z-axis, and then around the x'-axis.

Assume the 3D coordinates of a point, P, are known in the x-y-z coordinate reference frame: (x,y,z).
i) We would now like to know the coordinates of P in the x'-y'-z' coordinate reference frame, which is rotated θ radians around the z/z' axis (refer to the first diagram below).
ii) Furthermore, we would also like to know the coordinates of P in the x"-y"-z" coordinate reference frame, which is rotated γ radians around the x'/x" axis (refer to the second diagram below).

First Rotational Transformation

Referring to the diagram above, in the x-y-z coordinate reference frame, the point P has coordinates (x,y,z).
In the x'-y'-z' coordinate reference frame, which is rotated θ radians around the z/z' axis, the coordinates are:
x' = x cos(θ) - y sin(θ)
y' = x sin(θ) + y cos(θ)
z' = z

Second Rotational Transformation

Referring to the diagram above, in the x'-y'-z' coordinate reference frame, the point P has coordinates (x',y',z').
In the x"-y"-z" coordinate reference frame, which is rotated γ radians around the x'/x" axis, the coordinates are:
x" = x'
y" = y' cos(γ) - z' sin(γ)
z" = y' sin(γ) + z' cos(γ)

The utility posted on this page accepts as inputs the (x, y, z) coordinates of P, as well as the two angles θ and γ. It then calculates and outputs the (x', y', z') and (x", y", z") values.

HOW TO USE THIS UTILITY
Enter the (x, y, z) coordinates of P. Also enter the values (in radians) for θ and γ
Click the "Solve" button.
The coordinates of P in the x'-y'-z' and x"-y"-z" coordinate reference frames are output below.