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Chapter 2
Nomenclature
The nomenclature for geometry and rotations employs the following conventions. A vector x is a
column matrix of three elements, measuring the vector relative to a particular basis (or axes, or frame).
The basis is indicated as follows:
a) x A is a vector measured in axes A;
b) x EF/A is a vector from point F to point E, measured in axes A.
A rotation matrix C is a three-by-three matrix that transforms vectors from one basis to another:
c) C BA transforms vectors from basis A to basis B, so x B = C BA x A .
The matrix C BA defines the orientation of basis B relative to basis A, so it also may be viewed as rotating
the axes from A to B. For a vector u, a cross-product matrix u is defined as follows:
⎡
u = ⎣ 0 −u3
u3 0
⎤
u2
−u1 ⎦
−u2 u1 0
such that uv is equivalent to the vector cross-product u × v. The cross-product matrix enters the relation
between angular velocity and the time derivative of a rotation matrix:
˙C AB = −ω AB/A C AB = C AB ω BA/B
(the Poisson equations). For rotation by an angle α about the x, y,orz axis (1, 2, or 3 axis), the following
notation is used:
⎡
1
Xα = ⎣ 0
0
cos α
⎤
0
sin α ⎦
0 − sin α cos α
⎡
cos α
Yα = ⎣ 0
0
1
⎤
− sin α
0 ⎦
sin α 0 cos α
⎡
cos α
Zα = ⎣ − sin α
sin α
cos α
⎤
0
0 ⎦
0 0 1
Thus for example, C BA = XφYθZψ means that the axes B are located relative to the axes A by first
rotating by angle ψ about the z-axis, then by angle θ about the y-axis, and finally by angle φ about the
x-axis.