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and consider the frame F = ( u 1 , u 2 , u 3 ). The rotation R defined by F -1
then solves the
problem. The matrix for R is the same one as we got before, namely,
2
13
18
713
3
7
Ê
ˆ
-
Á
Á
Á
Á
Á
˜
˜
˜
˜
˜
13
7
6
7
0
.
3
13
12
713
2
7
-
-
Ë
¯
Actually, the fact that we got the same answer is accidental since the problem is under-
constrained and there are many rotations that rotate X to the x-y plane.
2.5.2.2. Example. To find the rotation R which rotates the plane X defined by the
equation y - z = 0 to the x-y plane.
Solution.
By inspection it is clear that the vectors
1
2
1
2
1
2
1
2
= Ê
Ë
ˆ
¯
Ê
Ë
ˆ
¯
= (
)
u
100
,, ,
u
1
,
,
,
and
u
=-
0
,
,
1
2
3
are an orthonormal basis of R 3 with u 1 and u 2 a basis for X . See Figure 2.30. Define
the orthogonal matrix A by
Ê
ˆ
10
0
Á
Á
Á
Á
Á
˜
˜
˜
˜
˜
1
2
1
2
= (
) =
TTT
A
uuu
123
0
-
.
1
2
1
2
0
Ë
¯
A is matrix for the rotation R we are looking for. It is easy to check that u i A = e i . Note
that
z
y
u 2
u 3
x
u 1
Figure 2.30.
Example 2.5.2.2.
 
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