determine a, b, c so that A is orthogonal where A= 0 2b c a b -c a b c
Answers
Answer:
Step-by-step explanation:
Let A=
⎣
⎢
⎢
⎡
0
a
a
2b
b
−b
c
−c
c
⎦
⎥
⎥
⎤
and A
T
=
⎣
⎢
⎢
⎡
0
2b
c
a
b
−c
a
−b
c
⎦
⎥
⎥
⎤
As A is orthogonal ∴AA
T
=I
⇒
⎣
⎢
⎢
⎡
0
a
a
2b
b
−b
c
−c
c
⎦
⎥
⎥
⎤
⎣
⎢
⎢
⎡
0
2b
c
a
b
−c
a
−b
c
⎦
⎥
⎥
⎤
=
⎣
⎢
⎢
⎡
1
0
0
0
1
0
0
0
1
⎦
⎥
⎥
⎤
=
⎣
⎢
⎢
⎡
4b
2
+c
2
2b
2
−c
2
−2b
2
+c
2
2b
2
−c
2
a
2
+b
2
+c
2
a
2
−b
2
−c
2
−2b
2
+c
2
a
2
−b
2
−c
2
a
2
+b
2
+c
2
⎦
⎥
⎥
⎤
=
⎣
⎢
⎢
⎡
1
0
0
0
1
0
0
0
1
⎦
⎥
⎥
⎤
∴ Using definition of equality of two matrices
4b
2
+c
2
=1,2b
2
−c
2
=0,a
2
+b
2
+c
2
=1
On solving them, a=±
2
1
,b=±
6
1
,c=±
3
1
Let A=
⎣
⎢
⎢
⎡
0
a
a
2b
b
−b
c
−c
c
⎦
⎥
⎥
⎤
and A
T
=
⎣
⎢
⎢
⎡
0
2b
c
a
b
−c
a
−b
c
⎦
⎥
⎥
⎤
As A is orthogonal ∴AA
T
=I
⇒
⎣
⎢
⎢
⎡
0
a
a
2b
b
−b
c
−c
c
⎦
⎥
⎥
⎤
⎣
⎢
⎢
⎡
0
2b
c
a
b
−c
a
−b
c
⎦
⎥
⎥
⎤
=
⎣
⎢
⎢
⎡
1
0
0
0
1
0
0
0
1
⎦
⎥
⎥
⎤
=
⎣
⎢
⎢
⎡
4b
2
+c
2
2b
2
−c
2
−2b
2
+c
2
2b
2
−c
2
a
2
+b
2
+c
2
a
2
−b
2
−c
2
−2b
2
+c
2
a
2
−b
2
−c
2
a
2
+b
2
+c
2
⎦
⎥
⎥
⎤
=
⎣
⎢
⎢
⎡
1
0
0
0
1
0
0
0
1
⎦
⎥
⎥
⎤
∴ Using definition of equality of two matrices
4b
2
+c
2
=1,2b
2
−c
2
=0,a
2
+b
2
+c
2
=1
On solving them, a=±
2
1
,b=±
6
1
,c=±
3
1