)
Two important properties of scalar product
Property 1 (Distributivity of scalar product over addition) Let ,   and  
a b
c
rr
r  be
any three vectors, then
(
)
a
b
c
⋅
r+
r
r  =
 
 
a b
a c
⋅
+
⋅
rr
r r
Fig 10 19
© NCERT
not to be republished
VECTOR ALGEBRA
443
(i)
B
C
A
l
B
l
A
C
(ii)
A
B
C
l
(iv)
l
C
B
A
(iii)
θ
θ
θ
θ
p
p
p
p
a
a
a
a
(90 <
< 180 )
0
0
θ
(0 <
< 90 )
0
0
θ
(270 <
< 360 )
0
0
θ
(180 <
< 270 )
0
0
θ
Property 2 Let  
aand 
rb
r
 be any two vectors, and λ be any scalar 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