# Rotationmatrix for coordinate system

this is my first question with regard to mathematics. So, if I made any mistake in terms of naming and/or convention, please let me know.

I'm having a little problem with rotating a coordinate system in 3D space. I have 5 points in 3D space and two coordinate systems, one for the world and one local on the object defined by said 5 points. Both coordinate systems are identical at the beginning.

To represent these coordinate systems, I'm using 3x3 matrices. They are initialized with a 3x3 identity matrix. To rotate these points, I'm using 3x3 rotation matrices.

The problem I have is this:

When I rotate the object defined by 5 points for 45° around any axis, let's say the X axis, the object local coordinate system rotates as well. Now I want to rotate this object again, this time around the Y axis around the objects local coordinate system.

I know how to rotate it around the world coordinate system, by just concatenating the rotations, if I'm not mistaken at least. But I'm not sure how to rotate around the objects local coordinate system.

An idea I had: When I want to create a rotation matrix for the object local coordination system, I can create a rotation matrix for the world matrix and apply the same rotations to it as I did to the objects local coordination system.

However, I'm not sure if that will give me correct results. Can you guys give me any hints?