7—Operators and Matrices 155Here the notationR~θrepresents the function prescribing a rotation byθabout the axis pointing along
ˆθ. These equations are the same as Eqs. (7.11) and (7.12).
The corresponding equations for the other two rotations are now easy to write down:Rβ~e 2
(
~e 1
)
=~e 1 cosβ−~e 3 sinβ, Rβ~e 2
(
~e 2
)
=~e 2 , Rβ~e 2
(
~e 3
)
=~e 1 sinβ+~e 3 cosβ (7.33)
Rγ~e 3
(
~e 1
)
=~e 1 cosγ+~e 2 sinγ, Rγ~e 3
(
~e 2
)
=−~e 1 sinγ+~e 2 cosγ, Rγ~e 3
(
~e 3
)
=~e 3 (7.34)
From these vector equations you immediate read the columns of the matrices of the components of the
operators as in Eq. (7.6).
(Rα~e 1
) (
Rβ~e 2
) (
Rγ~e 3
)
1 0 0
0 cosα −sinα
0 sinα cosα
,
cosβ 0 sinβ
0 1 0
−sinβ 0 cosβ
,
cosγ −sinγ 0
sinγ cosγ 0
0 0 1
(7.35)
As a check on the algebra, did you see if the rotated basis vectors from any of the three sets of equations
(7.32)-(7.34) are still orthogonal sets?
Do these rotation operations commute? No. Try the case of two 90 ◦rotations to see. Rotate
by this angle about thex-axis then by the same angle about they-axis.
(
R~e 2 π/ 2
)(
R~e 1 π/ 2
)
=
0 0 1
0 1 0
−1 0 0
1 0 0
0 0 − 1
0 1 0
=
0 1 0
0 0 − 1
−1 0 0
(7.36)
In the reverse order, for which the rotation about they-axis is done first, these are
(
R~e 1 π/ 2
)(
R~e 2 π/ 2
)
=
1 0 0
0 0 − 1
0 1 0
0 0 1
0 1 0
−1 0 0
=
0 0 1
1 0 0
0 1 0
(7.37)
Translate these operations into the movement of a physical object. Take the samex-y-zcoor-
dinate system as in this section, withxpointing toward you,yto your right andzup. Pick up a book
with the cover toward you so that you can read it. Now do the operationR~e 1 π/ 2 on it so that the cover
still faces you but the top is to your left. Next doR~e 2 π/ 2 and the book is face down with the top still
to your left. See problem7.57for and algebraic version of this.
Start over with the cover toward you as before and doR~e 2 π/ 2 so that the top is toward you
and the face is down. Now do the other operationR~e 1 π/ 2 and the top is toward you with the cover
facing right — a different result. Do these physical results agree with the matrix products of the last
two equations? For example, what happens to the vector sticking out of the cover, initially the column
matrix( 1 0 0 )? This is something that you cannot simplyread. You have to do the experiment for
yourself.
7.7 Areas, Volumes, Determinants
In the two-dimensional example of arrows in the plane, look what happens to areas when an operator
acts. The unit square with corners at the origin and(0,1),(1,1), 1 ,0)gets distorted into a paral-
lelogram. The arrows from the origin to every point in the square become arrows that fill out the
parallelogram.