Evaluate
the quantity 
,  if  |
| 1, |
| 4  and |
| 2
a b
b c
c a
a
b
c
μ =
⋅
+
⋅
+ ⋅
=
=
=
r
r
r
r
r
r r
r
r Solution Since 
0
a
b
c
+
+
=
r
r
r
r
, we have
(
)
a
a
b
c
 
 
r
r
r
r  = 0
or
a a
a b
a c
⋅
+
⋅
+
⋅
r
r
r r
r
r
 = 0
Therefore
a b
a c
⋅
+
⋅
r
r
r
r
 =
2
1
−a
= −
r (1)
Again,
(
)
b
a
b
c
⋅
+
+
r
r
r
r  = 0
or
a b
b c
⋅
+
⋅
r
r r
r
 =
2
16
−b
= −
r (2)
Similarly
a c
b c
⋅
+
r⋅
r r
r  = – 4