(^62) Kinematics and Dynamics of a Point Mass
Fig. 2.1.9 3-D Rotation of Frames
Similarly, we take the dot product of both sides of (2.1.33) with j:
V = xi(ii • 3) + 2/i(Ji • 3) + i(«i • 3) (2.1.35)
and with k:
z = xi{i\ • k) + j/i& • k) + zi(ki • k). (2.1.36)
The three equations (2.1.34)-(2.1.36) can be written as a single matrix -
vector equation,
i • 3 3-3 K-i-3
Kii • k 3i ' £ ki • k
(2.1.37)
Consider the matrix
i 'l 3i ' % Ki •
U = I 5i • 3 3i3 Ki • 3
Ji • k j 1 -k KI-K,
(2.1.38)
EXERCISE 2.1.16. (a)c Verify that the matrix U is orthogonal, that
is UUT = UTU = I. Hint: 1 = h • fc = (f • d)^2 + (i • ti)^2 + (A • i"i)^2 ,
0 = i • Ji = (i • )(ii • «) + (i • i)(ji • i) + (ii • K)C?I • «)• WA Veri/j/ i/iai
£/ie determinant of the matrix U is equal to 1. (c)A Verify that the matrix
U is a representation, in the basis (i\, J1; k), of an orthogonal transfor-
mation (see Exercise 8.1.4, page 453, in Appendix). This transformation
rotates the frame 0\ so that {i\, j\, «i) moves into (i, 3, k). (d)A Verify
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