# 3-D Rotation Matrices

When rotating a 3-dimensional coordinate system, we apply three rotation matrices $$R_x, R_y,$$ and $$R_z$$ which are to be multiplied by the primary point $$(x,y,z)$$ to get the new coordinates $$(x',y',z')$$. Does the order of the matrices matter? And how are they applied? (Say, $$(x',y',z')=R_xR_yR_z(x,y,z)$$.)

Thanks!

• Some of the ways to define a rotation involve three matrices like that. There are other ways to construct a rotation. If you use $R_x,$ $R_y,$ and $R_z$ then yes, the order matters, no, there is not a single "standard" way to do it. That's why you can go on some Web sites that explain Euler angles and be told there are $24$ different variations of the method. – David K Feb 12 at 18:16
• It seems silly to picture this as multiplying by three matrices. You can multiply the three matrices together into a single matrix, one which will do the right thing for any point you supply it. Yes, the order of those three matrices will matter. – rschwieb Feb 12 at 19:23
• Famously, the noncommutativity of 3D rotation can be described with quaternions. – J.G. Feb 12 at 19:33
• you might be interested to read this related post – G Cab Feb 12 at 19:37
• Try it out with some objects in front of you. Rotate around one axis and then another. Then try doing the same rotations in the opposite order. – Doug M Feb 12 at 19:48