Scalar product or dot product:
The scalar product of two vectors 
and 
is defined as the product of the magnitudes of and and the cosine of the angle between them.
If A and B creates angle θ then, . = AB
.
Properties of dot product:
(i) . = BA = AB = .
(ii) If the angle between and is 
then, . = ABcos = AB
(iii) . = AAcos = 
(iv) If the angle between and is 
then, . = ABcos = 0
(v) 
. = 
. = 
. = 1.1cos = 1
(vi) . = . = . = 1.1cos = 0
Concepts on dot product:
1. Work done: A force 
= (2î + 3ĵ – k̂) N is applied on a body to move it from point P (1, 2, -1) m to point Q (2, 4, 1) m. What is the work done required?
Force = (2î + 3ĵ – k̂) N and displacement of the body 
= (2-1 ) î + (4 – 2)ĵ + (1 + 1)k̂ = (î + 2ĵ + 2k̂) m.
Work done W = . = (2î + 3ĵ – k̂).(î + 2ĵ + 2k̂) = 2 + 6 – 2 = 6J.
2. Projection: Projection of 
on 
or component of on . PQ represents the magnitude of 
. PR and QS are two perpendiculars drawn on and PT is parallel to RS. The component of on is RS = PT = PQcos
[where is the angle between and ] .
= A.1.cos = .
EXAMPLE: What is the component of = 2î + 3ĵ + k̂ on = î + 2ĵ + 2k̂?
Component of on = . = 
= 
= 
.
3. Angle between two vectors: If is the angle between and then = 
.
Example: What is the angle between = 2î + 3ĵ + k̂ and = î + 2ĵ – 2k̂?
= 
= 
= 
.
4. Perpendicular vectors: If and are perpendicular to each other then, . = 0.
Example: What is the value of n so that = 2î – 3ĵ + k̂ and = î + nĵ + 2k̂ are perpendicular to each other?
Here (2î – 3ĵ + k̂).( î + nĵ + 2k̂) = 0
Or, 2 – 3n +2 = 0
n = 
.