• L = rotation matrix (or matrix of direction cosines)
• lij = cos of angle between ith axis of S ′ and jth axis of S
This is the fundamental result for the rotation of a vector. Note that by this we
mean that the vector is not rotated – it stays fixed in space – but that the basis is
rotated. (More on this point later.)
For example, consider the rotation through angle α about the z-axis. (NB anticlocky
y′
x′
α
x
z, z ′
wise when looking at tip of arrow.) Then we have


cos α sin α 0
Lz (α) =  − sin α cos α 0  ,
0
0 1
(~e′x · ~ex = cos α, ~e′x · ~ey = cos(π/2 − α) = sin α, ~e′y · ~ex = cos(π/2 + α) = − sin α, and
~e′y · ~ey = cos α etc..)
Similarly (by direct evaluation or cyclic rotations of axes of previous result)




1
0
0
cos γ 0 − sin γ
cos β sin β  ,
0 1
0 .
Lx (β) =  0
Ly (γ) = 
0 − sin β cos β
sin γ 0
cos γ
Note that as Lz (−α)Lz (α) = I and LTz (α) = Lz (−α) (by inspection) then we have
LTz (α)Lz (α) = I ie Lz is an orthogonal matrix.
The general result follows because the length of the vector is invariant,
′2
ak = (lki ai )(lkj aj )
2
~a =
,
a2i = δij ai aj
and as this is true for all ai then LT L = I, so L is an orthogonal matrix and hence
has determinant ±1 and hence an inverse always exists. Thus we have
LT L = I
or LT = L−1
17
or LLT = I ,
Alternatively we may consider the basis when we have a′i~e′i = ai~ei , which means that
lij aj ~e′i = aj ~ej which is true for all aj so lij e~i ′ = ~ej or lkj lij ~e′i = lkj ~ej or
~e′i = lij ~ej .
A matrix representing a rotation is orthogonal and has determinant +1. These are
called proper transformations.
[det L is +1 rather than −1 as for infinitesimal transformations we must have that
L → I continuously.]
But if (~ex , ~ey , ~ez ) is right handed (RH) and (~e′x , ~e′y , ~e′z ) is left handed (LH) then
det L = −1.
For example a reflection in the origin
y
z′
x′
x
z
y′
where


−1
0
0
0 ,
L =  0 −1
0
0 −1
gives det L = −1. These are called improper transformations.
2.1.1
Composition of two rotations
Let ~e′i = l1ij ~ej and ~e′′i = l2ij ~e′j , then we have
~e′′i = (L2 L1 )ij ~ej ,
(note the reversed order).
For example if


0 1 0
L1 =  −1 0 0  ,
0 0 1


0 0 −1
0 ,
L2 =  0 1
1 0
0
18
so L1 represents a rotation about Oz through π/2, while L2 represents a rotation
about Oy ′ again of π/2.
For L2 L1 :
y
S
x′
S′
z ′′
S ′′
x′′
x
y′
y ′′
z′
z


0 0 −1
0 
L2 L1 =  −1 0
0 1
0
~ex′′ = −~ez
~ey′′ = −~ex .
~ez ′′ = ~ey
or
Alternatively L2 followed by L1 gives:
S
y
S′
y′
S ′′
x′′
x′
x
z′
z
z ′′
y ′′


0 1 0
L1 L2 =  0 0 1 
1 0 0
or
~ex′′ = ~ey
~ey′′ = ~ez .
~ez ′′ = ~ex
So L2 L1 6= L1 L2 , ie rotations are non-commutative.
Three rotations first about z-axis (α), then y ′ -axis (β) then z ′′ -axis (γ) give a general
rotation – Euler angles:
L(α, β, γ) = Lz ′′ (γ)Ly′ (β)Lz (α) .
(Note that conventions can vary.) Expanding gives:



cos γ sin γ 0
cos β 0 − sin β
cos α sin α
0 1
0   − sin α cos α
L =  − sin γ cos γ 0  
0
0 1
sin β 0
cos β
0
0

cos β cos α cos γ − sin α sin γ
cos β sin α cos γ + cos α sin γ
=  − cos β cos α sin γ − sin α cos γ − cos β sin α sin γ + cos α cos γ
sin β cos α
sin β sin α

0
0 
1

− sin β cos γ
sin β sin γ  .
cos β
(Nice picture given in Mathews and Walker p405.) A rather complicated result.
19
2.2
Rotation of Vectors
The previous transformations are called passive transformations (ie rotate basis,
keeping the vector fixed).
Alternatively we can keep the basis fixed and rotate the vector – active transformations. We now consider this.
2.2.1
General Rotation
Q
θ
S
P
~n × ~x
Q
θ
~y
S
T
P
~x
~n
O
O is a fixed point in the body, which rotates about an axis through O in the direction
of the unit vector ~n.
OS is the component of ~x in the ~n-direction, ie (~x · ~n)~n. Also ~n × ~x is parallel to
T~Q.
~ + ST
~ + T~Q
~y = OS
(~x − (~x · ~n)~n)
~n × ~x
= (~x · ~n)~n + SQ cos θ
+ SQ sin θ
| {z } |
|
{z
}
SP
|~n × ~x|
{z
}
| {z }
~ |
|ST
|T~Q|
c
~
SP
= (~x · ~n)~n + cos θ (~x − (~x · ~n)~n) + sin θ~n × ~x ,
(as SP = SQ and |~n × ~x| = SP = SQ).
This gives finally the important result
~y = ~x cos θ + (1 − cos θ)(~n · ~x)~n + (~n × ~x) sin θ .
20
d
T~Q