Chemistry Reference
In-Depth Information
Fig. 1.2
Counterclockwise
rotation of the point
P
1
by an
angle
α
in the
xy
plane
in the opposite sense to that of the rotation of the hands on his watch. A synonym
for counterclockwise here is right-handed. If the reader orients his/her thumb in
the direction of the rotational pole, the palm of his/her right hand will indicate the
counterclockwise direction. The transformation can be obtained as follows. Let r be
the length of the radius-vector,
r
, from the origin to the point
P
1
, and let
φ
1
be the
angular coordinate of the point measured in the horizontal plane starting from the
x
-direction, as shown in Fig.
1.2
. The coordinates of
P
1
are then given by
x
1
=
r cos
φ
1
y
1
=
rsin
φ
1
(1.3)
z
1
=
0
Rotating the point will not change its distance from the origin, but the angular co-
ordinate will increase by
α
. The angular coordinate of
P
2
will thus be given by
φ
2
=
α
. The coordinates of the image point in terms of the coordinates of the
original point are thus given by
φ
1
+
x
2
=
r cos
φ
2
=
r cos
(φ
1
+
α)
=
r cos
φ
1
cos
α
−
rsin
φ
1
sin
α
=
x
1
cos
α
−
y
1
sin
α
y
2
=
rsin
φ
2
=
rsin
(φ
1
+
α)
(1.4)
=
r cos
φ
1
sin
α
+
rsin
φ
1
cos
α
=
x
1
sin
α
+
y
1
cos
α
z
2
=
0
In this way the coordinates of
P
2
are obtained as functions of the coordinates of
P
1
and the rotation angle. This derivation depends simply on the trigonometric rela-
tionships for sums and differences of angles. We may also express this result in the
form of a matrix transformation. For this, we put the coordinates in a column vector
