Let a and b be two non-zero vectors represented by OA and OB respectively. The angle between a and b is defined to be the angle between OA and OB, i.e. �
AOB.

Note that 0� �
� AOB �
180�.

efinition of Scalar Product

The scalar product of two non-zero vectors
a and b, denoted by a.b, is defined as

�

a.b = |a||b|cos q,�

where q = angle between a and b.�

mportant Properties

a.b = 0 � a ^
b (if a � 0, b
� 0)

|a.b| = |a||b| �
a // b (if a � 0,
b � 0)

a.b = b.a

a.(b + c) = a.b + a.c

(la).b = l(a.b)
= a.(lb) where l
� R.

a.a = |a|^{2}

: a.b = a.c
does not imply b = c.

For the 3 mutually perpendicular unit vectors
i, j, k, we have
�

i.i = j.j = k.k = 1

i.j = i.k = j.k = 0

calar Product in Cartesian Form

Let a = a_{1}i + a_{2}j
+ a_{3}k, b = b_{1}i
+ b_{2}j + b_{3}k.� Then,
using the properties of scalar product, we have