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In-Depth Information
2
13
0
3
Ê
ˆ
Á
Á
Á
Á
˜
˜
˜
˜
13
010
3
13
.
2
13
-
0
Ë
¯
It follows that
9
13
6
13
Ê
Ë
ˆ
¯
()
=
BRB
1
=
,,
1
-
.
1
Let R
2
be the rotation about the x-axis through an angle q
2
where
13
7
6
7
cos
q
=
and
sin
q
=
.
2
2
R
2
will move
B
1
to the x-y plane and leave
A
1
fixed. Finally, R = R
2
R
1
will be the rota-
tion we are looking for because R leaves the origin fixed and maps the points
A
and
B
to the x-y plane. The matrix for R
2
is
10
0
Ê
ˆ
13
7
6
7
Á
Á
Á
˜
˜
˜
0
,
6
7
13
7
0
-
Ë
¯
and so the matrix for R is
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
¯
Before leaving this problem, let us look at another possible question. What is the
equation for the plane
X
1
= R
1
(
X
)? Note that, by definition, (x
1
,y
1
,z
1
) belongs to
X
1
if and only if R
-1
(x
1
,y
1
,z
1
) belongs to
X
. Therefore, since the matrix for
R
1
-1
is
2
13
0
3
Ê
ˆ
-
Á
Á
Á
Á
˜
˜
˜
˜
13
01 0
3
13
2
13
0
Ë
¯
