1
Rotation Matrices in Three-Dimensional Cartesian Coordinates
Instructor: Nam Sun Wang
rotate.mcd
The three most fundamental Rotation matrices are listed below. We can derive everything else from
these three.
Rotation about the x-axis:
1
R x ( φ)
0
Rotation about the y-axis:
cos( ψ ) 0 sin( ψ )
0
0 cos( φ)
Rotation about the z-axis
sin( φ)
R y( ψ )
0
0 sin( φ) cos( φ)
1
0
sin( ψ ) 0 cos( ψ )
cos( θ )
R z( θ )
sin( θ ) 0
sin( θ ) cos( θ ) 0
0
0
1
It matters which way we rotate first. For example, RxRz≠RzRx
cos( θ )
R x ( φ) . R z( θ )
sin( θ )
0
cos( φ) . sin( θ ) cos( φ) . cos( θ )
sin( φ)
sin( φ) . sin( θ ) sin( φ) . cos( θ ) cos( φ)
cos( θ )
R z( θ ) . R x ( φ)
cos( φ) . sin( θ )
sin( φ) . sin( θ )
sin( θ ) cos( φ) . cos( θ )
sin( φ) . cos( θ )
sin( φ)
0
cos( φ)
Successive Rotation. For example, rotate about the z axis first followed by rotation about the x-axis
results in a vector Rx*Rz*v, where v is the original vector before rotation. We let Mathcad construct
the six double rotation matrices with SmartMath, and we copy the results into the expressions
below. Notation: Rxz=RxRz
R xy( φ, ψ )
cos( ψ )
0
sin( ψ )
sin( φ) . sin( ψ )
cos( φ)
sin( φ) . cos( ψ )
cos( φ) . sin( ψ ) sin( φ) cos( φ) . cos( ψ )
cos( θ )
R xz( φ, θ )
sin( θ )
0
cos( φ) . sin( θ ) cos( φ) . cos( θ )
sin( φ)
sin( φ) . sin( θ ) sin( φ) . cos( θ ) cos( φ)
cos( ψ ) sin( φ) . sin( ψ ) cos( φ) . sin( ψ )
R yx( φ, ψ )
cos( φ)
0
sin( φ)
sin( ψ ) sin( φ) . cos( ψ ) cos( φ) . cos( ψ )
cos( ψ ) . cos( θ )
R yz( ψ , θ )
cos( ψ ) . sin( θ ) sin( ψ )
sin( θ )
cos( θ )
0
sin( ψ ) . cos( θ )
sin( ψ ) . sin( θ )
cos( ψ )
← Commonly used 3-D rotation matrix.
cos( θ )
R zx( φ, θ )
2
sin( φ) . sin( θ )
cos( φ) . sin( θ )
sin( θ ) cos( φ) . cos( θ )
cos( ψ ) . cos( θ )
R zy( ψ , θ )
sin( φ) . cos( θ )
sin( φ)
0
rotate.mcd
cos( φ)
sin( θ ) sin( ψ ) . cos( θ )
cos( ψ ) . sin( θ ) cos( θ ) sin( ψ ) . sin( θ )
sin( ψ )
cos( ψ )
0
Check: -R(•) is not reverse rotation or inverse.
1
R x ( φ ) . R x ( φ)
0
0
cos( φ)
0
0
2
sin( φ)
2
2 . cos( φ) . sin( φ)
2 . cos( φ) . sin( φ)
cos( φ)
2
sin( φ)
2
Check: Rotation forward followed by rotation backward by the same angle should give Identify.
R(-•)=R-1(•)
1
R x ( φ ) . R x ( φ)
0
0 cos( φ)
0
2
0
sin( φ)
2
0
cos( φ)
0
2
sin( φ)
2
Another check on R(-•)=R-1(•):
cos( ψ )
R xy( φ, ψ ) . R yx( φ, ψ )
2
sin( ψ )
2
0
2
sin( φ) . sin( ψ )
0
2
sin( φ) . sin( ψ ) . cos( φ)
0
2
cos( φ)
2
2
2
sin( φ) . cos( ψ )
cos( φ) . sin( φ)
sin( φ) . sin(
2
sin( φ) . cos( ψ ) . cos( φ)
co
Copy and paste the RHS:
cos( ψ )
2
sin( ψ )
2
0
2
sin( φ) . sin( ψ )
0
0
2
sin( φ) . sin( ψ ) . cos( φ)
|Symbolic|Simplify| gives:
1 0 0
0 1 0
0 0 1
Thus, R(•)*R(-•)=1
2
cos( φ)
0
2
2
sin( φ) . cos( ψ )
cos( φ) . sin( φ)
2
2
sin( φ) . cos( ψ ) . cos( φ)
2
sin( φ) . sin( ψ ) . cos( φ)
cos( φ) . sin( φ
2
2
cos( φ) . sin( ψ )
sin( φ)
2
3
rotate.mcd
Check: Rotation matrices are orthogonal. R(•)*RT(•)=RT(•)*R(•)=1, or equivalently, RT(•)=R-1(•)
1
R x ( φ ) . R x ( φ)
T
0
0 cos( φ)
2
0
0
sin( φ)
2
0
cos( φ)
0
2
Copy and paste the RHS:
1
0 cos( φ)
0
sin( φ)
2
cos( φ)
0
2
|Symbolic|Simplify| gives:
0
2
sin( φ)
0
1 0 0
0
0 1 0
2
sin( φ)
2
Thus, R(•)*RT(•)=1
0 0 1
Now, test the converse to see if RT(•)*R(•)=1.
1
T
R x ( φ) . R x ( φ)
0
0 cos( φ)
2
0
0
sin( φ)
2
0
cos( φ)
0
Copy and paste the RHS:
1
0 cos( φ)
0
sin( φ)
2
cos( φ)
0
sin( φ)
2
|Symbolic|Simplify| gives:
0
2
2
2
0
1 0 0
0
0 1 0
sin( φ)
2
Thus, RT(•)*R(•)=1
0 0 1
Another check on R xy:
cos( ψ )
R xy( φ, ψ ) . R xy( φ, ψ )
T
2
sin( ψ )
2
0
2
2
sin( φ) . sin( ψ )
0
2
sin( φ) . sin( ψ ) . cos( φ)
0
cos( φ)
2
2
2
sin( φ) . cos( ψ )
cos( φ) . sin( φ)
sin( φ) . si
2
sin( φ) . cos( ψ ) . cos( φ)
Copy and paste the RHS:
cos( ψ )
2
sin( ψ )
2
0
2
sin( φ) . sin( ψ )
0
0
2
sin( φ) . sin( ψ ) . cos( φ)
|Symbolic|Simplify| gives:
1 0 0
0 1 0
0 0 1
2
cos( φ)
0
2
2
sin( φ) . cos( ψ )
cos( φ) . sin( φ)
2
2
sin( φ) . cos( ψ ) . cos( φ)
2
sin( φ) . sin( ψ ) . cos( φ)
cos( φ) . sin(
2
2
cos( φ) . sin( ψ )
sin( φ)
2
4
rotate.mcd
Check: Rotation Matrices are normalized.
R x( θ )
cos( θ )
2
sin( θ )
2
which simplifies to unity.
Another example:
R xz( θ , φ)
2
2
cos( φ) . cos( θ )
2
2
sin( θ ) . cos( φ)
2
2
sin( φ) . cos( θ )
2
2
sin( φ) . sin( θ )
Copy and paste the RHS:
2
2
cos( φ) . cos( θ )
2
2
sin( θ ) . cos( φ)
2
2
sin( φ) . cos( θ )
2
2
sin( φ) . sin( θ )
|Symbolic|Simplify| gives:
1
We can generate a series of combinations of three successive rotations. The following is one
example:
cos( ψ ) . cos( θ )
R x ( φ) . R y ( ψ ) . R z( θ )
sin( φ) . sin( ψ ) . cos( θ )
cos( φ) . sin( ψ ) . cos( θ )
cos( φ) . sin( θ )
sin( φ) . sin( ψ ) . cos( θ )
cos( φ) . sin( ψ ) . cos( θ )
sin( ψ )
sin( φ) . sin( ψ ) . sin( θ )
cos( φ) . cos( θ )
sin( φ) . cos( ψ )
sin( φ) . sin( θ ) cos( φ) . sin( ψ ) . sin( θ )
sin( φ) . cos( θ )
cos( φ) . cos( ψ )
cos( θ ) . cos( ψ )
R xyz( φ, ψ , θ )
cos( ψ ) . sin( θ )
cos( φ) . sin( θ )
cos( ψ ) . sin( θ )
sin( ψ )
sin( φ) . sin( ψ ) . sin( θ )
cos( φ) . cos( θ )
sin( φ) . cos( ψ )
sin( φ) . sin( θ ) cos( φ) . sin( ψ ) . sin( θ )
sin( φ) . cos( θ )
cos( φ) . cos( ψ )