Figure 3.1 A vector observed in two different reference frames.
The components of
(3.5)
(3.6)
On the other hand, by recalling the definition of the basic column matrices from Chapter 1, the following equations can be written.
(3.7)
(3.8)
Hence, referring to Eq. (3.2),
(3.9)
(3.10)
3.2 Transformation Matrices Between Reference Frames
3.2.1 Definition and Usage of a Transformation Matrix
Consider two reference frames
In Eq. (3.11),
3.2.2 Basic Properties of a Transformation Matrix
1 (a) Inversion Property
Equation (3.11) can also be written in the following two ways: first by inverting
(3.12)
Equations (3.11)–(3.13) imply the following equalities.
(3.15)
1 (b) Orthonormality Property
As also mentioned in Chapter 1, all the reference frames considered in this book are assumed to be orthonormal, right‐handed, and equally scaled on their axes. Therefore, the magnitude of a vector
After substituting Eq. (3.11), Eq. (3.16) can be manipulated as shown below.
Equation (3.17) implies the following successive equations.
(3.18)
Furthermore, when Eqs.