� The scalar product of two given vectors 
aand
rb
r
 having angle θ between
them is defined as
|
||
| cos
a b
a b
⋅
=
θ
r
r
r
r Also, when a b
⋅
rr
 is given, the angle ‘θ’ between the vectors 
aand
rb
r
 may be
determined by
cosθ = |
||
a b|
a b
⋅
rr
r
r
� If θ is the angle between two vectors 
aand
rb
r
, then their cross product is
given as
a
b
×
r
r
=
ˆ
|
||
a b|sin
n
θ
r
r
where ˆn is a unit vector perpendicular to the plane containing 
aand
rb
r Such
that 
ˆ
, ,
r
ra b n form right handed system of coordinate axes � If we have two vectors 
aand
rb
r
, given in component form as
1
2
3 ˆ
ˆ
ˆ
a
a i
a j
a k
=
+
+
r
 and 
1
2
3 ˆ
ˆ
ˆ
b
b i
b j
b k
=
+
+
r
 and λ any scalar,
© NCERT
not to be republished
VECTOR ALGEBRA
461
then
a
b
+
r
r
 =
1
1
2
2
3
3 ˆ
ˆ
ˆ
(
)
(
)
(
)
a
b i
a
b
j
a
b
k
+
+
+
+
+
;
a
λr  =
1
2
3 ˆ
ˆ
ˆ
(
)
(
)
(
)
a i
a
j
a k
λ
+ λ
+ λ
;
rr