Then
(
a)
b
λ
⋅
r
r
 = (
)
(
)
(
)
a
b
a b
a
b
 
  
 
 
  
r
r
r
r
r
r
If two vectors 
aand 
rb
r
 are given in component form as 
1
2
3 ˆ
ˆ
ˆ
a i
a j
a k
+
+
 and
1
2
3 ˆ
ˆ
ˆ
b i
b j
b k
+
+
, then their scalar product is given as
a b
⋅
rr
 =
1
2
3
1
2
3
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
(
) (
)
a i
a j
a k
b i
b j
b k
+
+
⋅
+
+
=
1
1
2
3
2
1
2
3
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
(
)
(
)
a i
b i
b j
b k
a j
b i
b j
b k
⋅
+
+
+
⋅
+
+
 +
3
1
2
3
ˆ
ˆ
ˆ
ˆ
(
)
a k
b i
b j
b k
⋅
+
+
=
1 1
1 2
1 3
2 1
2 2
2 3
ˆ
ˆ
ˆ ˆ
ˆ ˆ
ˆ
ˆ ˆ
ˆ ˆ
ˆ
(
)
(
)
(
)
(
)
(
)
(
)
a b i i
a b i
j
a b i k
a b
j i
a b
j j
a b
j k
⋅
+
⋅
+
⋅
+
⋅
+
⋅
+
⋅
 + 
3 1
3 2
3 3
ˆ
ˆ
ˆ ˆ
ˆ
ˆ
(
)
(
)
(
)
a b k i
a b k j
a b k k
⋅
+
⋅
+
⋅
(Using the above Properties 1 and 2)
= a1b1 + a2b2 + a3b3
(Using Observation 5)
Thus
a b
⋅
rr
 =  
1 1
2 2
3 3
a b
a b
a b
+
+
10 6 2  Projection of a vector on a line
Suppose a vector AB
uuur
 makes an angle θ with a given directed line l (say), in the
anticlockwise direction  (Fig 10 20)