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0593_C02_fm Page 41 Monday, May 6, 2002 1:46 PM

Review of Vector Algebra 41

2.11 Reference Frames and Unit Vector Sets
In the analysis of dynamical systems, it is frequently useful to express kinematical and
dynamical quantities in several different reference frames. Orthogonal transformation
matrices (as discussed in this section) are useful in obtaining relationships between the
representations of the quantities in the different reference frames.
ˆ
To explore these ideas consider two unit vector sets n and and an arbitrary V as in
i n
i
Figure 2.11.1. Let the sets be inclined relative to each other as shown. Recall from
Eq. (2.6.19) that V may be expressed in terms of the n as:
i
V = ( V n n ) 1 +( Vn n ) 2 +( V n n ) 3 = ( Vn n ) i = Vn i (2.11.1)
⋅
⋅
⋅
⋅
i
1
3
2
i
where the V are the scalar components of V. Similarly, V may be expressed in terms of
i
the as:
ˆ
i n
V = ( V n n ) = V n ˆ (2.11.2)
ˆ
⋅ ˆˆ
i i i i
Given the relative inclination of the unit vector sets, our objective is to obtain relations
ˆ
between the V and the V . To that end, let S be a matrix with elements S defined as:
i i ij
⋅
S = nn ˆ (2.11.3)
ij i j
Consider the n : from Eq. (2.11.2), we can express n as:
i i
n = ( n n n ) = S n ˆ (2.11.4)
⋅ ˆˆ
1 1 i i 1i i
Similarly, n and n may be expressed as:
2 3
n
n = S ˆ and n = S ˆ i (2.11.5)
n
3i
i
2
3
2i
Thus, in general, we have:
n = S n ˆ (2.11.6)
i ij j
V
n ˆ
3
n
3
n ˆ
2
n
2
n ˆ
FIGURE 2.11.1 1
Two unit vector sets. n 1