Since va and vh are physically the same vector v, the magnitude of a must be the same as that of vb, that is, v2 is an invariant of the transformation. From (A.4,3) this requires

v2 = lvh = TaJhaLhay a = Tau (A.4,5)

It follows from the last equality of (A.4,5) that

KaUa = I (A.4,6)

Equation (A.4,6) is known as the orthogonality condition on L^r From (A.4,6) it fol­lows that and hence that |Lfe,| is never zero and the inverse of Lha always exists. In view of (A.4,6) we have, of course, that

I T _ І -1 _ t

*^ba *^ah

that is the inverse and the transpose are the same. Equation (A.4,6) together with (A.4,3b) yields a set of conditions on the direction cosines,

It follows from (A.4,8) that the columns of Lba are vectors that form an orthogonal set (hence the name “orthogonal matrix”) and that they are of unit length.

Since (A.4,8) is a set of six relations among the nine liJt then only three of them are independent. These three are an alternative to the three independent Euler angles for specifying the orientation of one frame relative to another.


The transformations associated with single rotations about the three coordinate axes are now given. In each case Fa represents the initial frame, Fh the frame after rota­tion, and the notation for L identifies the axis and the angle of the rotation (see Fig. A.4). Thus in each case

Уь = L,(X,)va

By inspection of the angles in Fig. A.4, the following matrices are readily verified.

’10 0 L^X,) = 0 cosX, sinX, 0 — sinXj cosX,

‘cosX2 0 — sinX2 L2(X2) =010

sin X2 0 cos X2

cosX3 sinX3 0 L3(X3) = -sinX3 cosX3 0 0 0 1

*»l *ai

Figure A.4 The three basic rotations, (a) About xav (b) About xar (c) About xay

The transformation matrix for any sequence of rotations can be constructed readily from the above basic formulas. For the case of Euler angles, which rotate frame FE into FB as defined in Sec. 4.4, the matrix corresponds to the sequence (X3, X2, Xt) = (iff, в, Ф), giving

L,(</>) • L2(0) • L3(ф)


[The sequence of angles in (A.4,11) is opposite that of the rotations, since each trans­formation matrix premultiplies the vector arrived at in the previous step.] The result of multiplying the three matrices is

cos в cos ф

cos в sin ф

— sin в

sin ф sin в cos ф — cos ф sin ф

sin ф sin в sin ф + cos ф cos ф

sin ф cos в


cos ф sin в COS Ф + sin ф sin ф

cos ф sin в sin ф —sin ф cos Ф

cos ф cos в