CE507 Lecture 2
Coordinate Transformation
Start with a coordinate system
1
2
3
(
,
,
)
x x
x
a right-hand system with origin O.
Rotate it to a new coordinate system,
1
2
3
(
,
,
)
x x
x
′
′
′
with the same origin O.
Let
,
,
1
2
3
ˆ ˆ
ˆ
e e e
be the unit vectors along the
1
2
3
,
,
x
x
x
−
−
−
axes,
and
,
,
1
2
3
ˆ ˆ
ˆ
e e e
′
′
′
the corresponding unit vectors along the
1
2
3
,
,
x
x
x
′
′
′
−
−
−
axes.
Define:
angle between the
- (new) axis and
- (old)axis.
ij
i
j
x
x
α
′
=
and
ij
a
= cos (
ij
α
) =
ˆ
ˆ
i
j
e
e
′⋅
, called the direction cosines
.
We will study the matrix
[ ]
(
)
ij
a
a
=
in this lecture.
Note that a given vector,
x
±
, say, can have two different representations w.r.t.
the two coordinate systems.

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