ME 6590 Multibody Dynamics
Coordinate Transformation (Rotation) Matrices
Relationships Between Unit Vectors in Different Reference Frames
The unit vectors of two different mutually perpendicular unit vector sets e e e
1 2 3
and n n n
1 2 3
) can be related using transformation matrices. To do this, we note that we can write e i
 j
3 

1
( i j j
for i

1,2,3 .
Or, in matrix form,
{ }
 e e
1
2

  



(
(
(
1


2 1 e n
1
1
)
)
)
(
(
(
2

 e n
2
2
2
)
)
) (
(
(
2


3
) n e n
3
3
)
)


  n
3
 where C ij e n i j e n i j
)



C
C
C
11
21
31
C
C
C
12
22
32
C
C
C
13
23
33

 n n
2
 
3

represents the cosine of the angle between the unit vectors e i
and n j
. The matrix C is called the direction cosine matrix. It can be shown that the matrix [ ] is orthogonal, so its inverse is equal to its transpose .
Using this last result, we can write { } [ ]{ } n

C
T e .
Relationships Between Vector Components in Different Reference Frames
Given the representations of a vector a in two different reference frames a
 i
3 

1
 i
3 

1
 i i
, the components a i

1,2,3) can be related to the i
( 1,2,3) using transformation matrices as follows. Writing the above equation in matrix form, we have a
T e a
T n a
T
C
T
{ } { } { } { } { } [ ] { }
1/2

Comparing both sides of the equation, we conclude
{ }
T 
{ } [ ]
T
or { } [ ]{ }
Finally, using the fact that [ ] is orthogonal, we conclude also that a C
T
{ } [ ] { } .
Dot and Cross Products Revisited
We can also use transformation matrices to take the dot or cross products of two vectors expressed in two different reference frames as follows
   
   
Recall that




 a
0
3
 a
3 a
0
 a
2 a
1

2 a
0
1




.
2/2