|
|
a
b
+
r
r
 = |
|
|
|
a
+b
r
r
,
then
| AC|
uuur
 = | AB|
uuur+| BC |
uuur
showing that the points A, B and C are collinear Example 21 Show that the points 
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
A( 2
3
5 ), B(
2
3 )
i
j
k
i
j
k
−
+
+
+
+
 and 
ˆ
C(7ˆ
)
i
k
−
are collinear Solution We have
 AB
uuur
 =
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
(1
2)
(2
3)
(3
5)
3
2
i
j
k
i
j
k
 
 
 
 
 
 
 
,
BC
uuur
 =
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
(7
1)
(0
2)
( 1
3)
6
2
4
i
j
k
i
j
k
 
 
 
 
 
 
 
 
 
,
AC
uuur
 =
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
(7
2)
(0
3)
( 1
5)
9
3
6
i
j
k
i
j
k
 
 
 
 
 
 
 
 
 
| AB|
uuur
 =
14, | BC|
 2 14  and  | AC| 3 14
 
uuur
uuur
Therefore
AC
uuur
 = | AB|
uuur+| BC |
uuur
Hence the points A, B and C are collinear �Note   In Example 21, one may note that although AB
BC
CA
0
+
+
=
uuur
uuur
uuur
r  but the
points A, B and C do not form the vertices of a triangle