Alternatively, 
1
AB
2CD
  
uuur
uuur
 which implies that AB and CD
uuur
uuur  are collinear vectors Example 28 Let 
,
a band
c
rr
r  be three vectors such that |
| 3, |
| 4, | | 5
a
b
c
=
=
=
r
r
r
 and
each one of them being perpendicular to the sum of the other two, find |
a b c|
+ +
r
r
r Solution Given 
(
)
a
b
c
⋅
r+
r
r  = 0,
(
)
0,
(
)
0 b
c
a
c
a
b
⋅
+
=
⋅
+
=
r
r
r
r
r
r
Now
2
|
|
a
b
c
+
+
r
r
r
 =
2
(
)
(
) (
)
a
b
c
a
b
c
a
b
c
+
+
=
+
+
⋅
+
+
r
r
r
r
r
r
r
r
r
=
(
)
(
)
a a
a
b
c
b b
b
a
c
⋅
+
⋅
+
+
⋅
+
⋅
+
r
r r
r
r r
r
r
r
r
+