Physics 325
Lecture 2
If we rotate first by
β
and then by
δ
, we get the following:
'
cos
sin
First rotate by
:
'
sin
cos
"'
cos
sin
Now rotate by
:
sin
cos
ββ
β
δδ
δ
−
⎛⎞
=
⎜⎟
⎝⎠
−
=
xx
yy
AA
cos
sin
cos
sin
sin
cos
sin
cos
cos
cos
sin
sin
cos
sin
sin
cos
cos
sin
sin
cos
cos
cos
sin
sin
cos
sin
sin
cos
δβ
βδ
−−
⎛ ⎞
==
⎜ ⎟
⎝ ⎠
−
+−
+− +⎛⎞
=
++
x
y
x
y
x
y
A
A
A
A
A
A
⎛
⎞
This is equivalent to a single rotation by
β+δ
, as you would expect.
Notice that in this
case it would not matter if we did the
rotation first and then the
rotation, or vice
versa.
The rotation matrices, in the 2 dimensional case, commute.
This is not so for
three dimensions and higher, as is nicely shown in Figure 1.9 in the text.
The rotation matrices have the property that their transpose is their inverse:
22
cos
sin
cos
sin
cos
sin
cos
sin
sin
cos
sin
cos
sin
cos
sin
cos
10
cos
sin
cos
sin
cos
sin
01
cos
sin
cos
sin
cos
sin
=
−
+
−+
T
−
This property insures that dot products are invariant under rotation:
()
( )
T
TT
T
A B
A
B
RA
RB
A
R R B
A B
A B
′′
=
GG
ii
T
For instance, the length of a vector (
G G
i
) does not change under rotation.
Cross Products
The cross product between two vectors yield a third vector, perpendicular to the first two
and with magnitude given by
sin
CA
BA
B
θ
=×=
G
G
G
(2.1)
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View Full Documentwhere
θ
is the angle between
and
A
G
B
G
.
Take for instance the three unit vectors:
Cross products between different combinations of these work as follows:
1
ˆ
e
3
ˆ
e
2
ˆ
e
(2.2)
12 323 131 2
21
332
113
ˆˆ ˆˆˆ ˆˆˆ ˆ
,,
ˆˆ
ˆˆˆ
ˆ
ee eee eee e
ee
eee
e
×=
−
−
−
2
We can summarize this by introducing a new, useful, item:
( )
ˆˆ ˆ
i
j
k
ijk
ee e
ε
i
where
( )
()
1 if
, ,
(1,2,3),(3,1,2),(2,3,1)
(1,3,2),(2,1,3),(3,2,1)
0 if
ijk
ijk
i
j or j
k or i
k
+=
⎧
⎪
=−
=
⎨
⎪
===
⎩
(2.3)
So there are many ways to evaluate the cross product.
You can use Equation 2.1 for the
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 Fall '08
 Staff
 Physics, mechanics, Derivative, Vector Calculus, Cos, Vector field, Gradient

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