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Vector transformation
and
,
Rotational matrix
,
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The relationship between
below:
Rotational matrix
Here we follow the approach of a mathematician
and define a vector as a set of three components that
transforms in the same manner as a displacement
when we change the coordinates. As always, the
displacement vector is the model for the behavior of
all vectors.
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So far we have used two different ways to describe
vector; (a) geometry approach---vector as an arrow,
(b) algebra approach--- vector as components of
Cartesian coordinates. However both approaches
are not very satisfactory and are rather naïve.
A more rigorous definition of vector starts with
the concept that the space is (a) isotopic, so no
preferred direction, (b) homogeneous, so no
preferred location. A physical quantity such as
displacement or force, should be independent of
the coordinate system we choose.
Let
1
2
Rotation about z-axis, using right-hand rule (Counter clockwise).
can be expressed
(2)
This is a rotation about the z-axis. In the text book, it
is a rotation about the x-axis. For rotation about x
axis is given on the next page:
This is rotation about x-axis, again we use right-hand rule, so the yaxis is rotated toward x-axis.
For the rotation about y-axis, using right-hand rule, we have:
Now we see that the matrix is a lot different from the previous
ones. In order to make the rotation matrix the same as in x and zaxis, we define the rotation about y-axis as clockwise, so the
rotational matrix for the y-axis will have similar form as other two
axes.
3
For rotation about y-axis, clockwise, the rotation matrix is given
Rotational matrix
Rotational matrix
(1)
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The above equations can be combined into one matrix notation;
4
Example 1
Rotational matrix
Rotational matrix
Now for the most general case, we can rotates the coordinate by
an arbitrary angle through an arbitrary direction, and the
rotation matrix is given below, however the determination of the
matrix elements will be a challenging task, since we only know
how to do it through the above three equations (1), (2) and (3)
We want to rotate the z-axis through an angle of
φ such that the axis is coincided with line
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(3)
Step #1
Rotates about the z-axis by an angle of
(4)
Step #2
Rotates about y’ axis, counterclockwise by an angle of θ
(5)
5
6
1
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Re-arrange eq. (5)
Problem 1.9
(6)
(7)
z
y
Rotational matrix
Rotational matrix
axis, by an angle of -φ
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Step #3
Rotates about the
Find the transformation matrix R that describes a rotation by 120°
about an axis from the origin through the point (1,1,1). The rotation
is clockwise.
x
By inspection, we can see that the transformation matrix R is given by
The final rotation matrix is
,
,
∙
∙
(9)
7
8
The rotation on page 8 is equivalent to:
1.
2.
Rotates about z-axis for -90°, and then
Then rotates about -axis clockwise 90°
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z
Rotational matrix
y
x
R=
0
0
1
0
0
0
0
0
1
0
1
0
1
0
0
0
0
1
0
1
0
0
0
0
1
0
0
1
°
°
φ
0
1
0
0
0
1
1
0
0
9
2