1.1.2 Change of basis

e3

−1 cos Q23 e′3

e′2

e2

e1′

e cos−1 Q 1 11

Consider a second right-handed orthonormal basis {e′i }. Since {ei } is a basis, eee123′′′,, can be expressed as a linear combination of eee123,,,

ee′iipp= Q . (1.1.28)

From (1.1.28),

Qij= ee′ i• j . (1.1.29)

The definition (1.1.10) with (1.1.29) shows that the Qij ’s are the direction cosines of the vectors e′i relative to the e j .

By orthonormality and (1.1.28),

δij==ee′′ i•• jQQQ ik e k e ′ j = ik jk . (1.1.30)

T Let the coefficients Qij be represented by a matrix Q . Then (1.1.30) shows that Q is the inverse matrix of Q , (i.e., Q is an orthogonal matrix),

QQTT = I = Q Q (1.1.31) which in component notation is,

QQik jk= δ ij= QQ ki kj . (1.1.32)

Premultiplication of (1.1.28) by Qij and use of (1.1.32) leads to the dual connections between the basis vectors,

ee′′iijj==QQ, e jiji e. (1.1.33)

October 15, 2007 1.1.2-1 From (1.1.22), (1.1.25) and (1.1.31),

(detQQQQQ)))2 === (detTT (det det 1, and hence

detQ =± 1. (1.1.34)

The situation in which detQ =+ 1 corresponds to maintenance of right-handedness of the basis vectors. In this case Q is proper orthogonal and may be interpreted as a rotation which takes {ei } into {e′i } . For a change of basis in which right-handedness is not preserved, detQ =− 1 and Q is improper orthogonal.

Let vvii, ′ be the components of a vector v with respect to bases {eeii },{′ }, respectively. Then, by use of (1.1.33),

ve===vvvQkk′′ j e j j kjk e ′.

Taking the dot product of this with e′i and applying (1.1.32),

vQviijj′′==, v jiji Qv, (1.1.35) which shows that the components of v transform under changes of orthonormal basis according to the same rule (1.1.33) applicable to the basis vectors themselves.

Example: Consider a change of basis for which ee′33= and,

eee112′ =+cosθθ sin ⎫ ⎬ , (1.1.36) eee′212=−sinθθ + cos ⎭

corresponding to a positive (i.e., anticlockwise) rotation through an angle θ about e3 . Then

⎡⎤cosθθ sin 0 Q =−⎢⎥sinθθ cos 0 (1.1.37) ⎢⎥ ⎢⎥001 ⎣⎦ and it can be seen that QQT = I and detQ = + 1.

October 15, 2007 1.1.2-2

Problem 1.1.3 Write down (a) the Q corresponding to a rotation θ about e2 , (b) the Q corresponding to a rotation φ about e1 , (c) the Q corresponding to (b) followed by (a).

Problem 1.1.4 Show that

⎡⎤cos 2θθ sin 2 0 Q ⎢⎥ =−⎢⎥sin 2θθ cos 2 0 ⎣⎦⎢⎥001 is an improper orthogonal matrix that represents a change of basis equivalent to a reflection in the plane through e3 inclined at a positive angle θ to e1 .

October 15, 2007 1.1.2-3