In this section, we summarize the properties of dot product as discussed above. Besides, some additional derived attributes are included for reference.

1: Dot product is commutative

This means that the dot product of vectors is not dependent on the sequence of vectors :

a
.
b
=
b
.
a
a
.
b
=
b
.
a

We must, however, be careful while writing sequence of dot product. For example, writing a sequence involving three vectors like a.b.c is incorrect. For, dot product of any two vectors is a scalar. As dot product is defined for two vectors (not one vector and one scalar), the resulting dot product of a scalar (a.b) and that of third vector c has no meaning.

2: Distributive property of dot product :

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

3: The dot product of a vector with itself is equal to the square of the magnitude of the vector.

a
.
a
=
a
x
a
cosθ
=
a
2
cos
0
°
=
a
2
a
.
a
=
a
x
a
cosθ
=
a
2
cos
0
°
=
a
2

4: The magnitude of dot product of two vectors can be obtained in either of the following manner :

a
.
b
=
a
b
cos
θ
a
.
b
=
a
b
cos
θ
=
a
x
(
b
cos
θ
)
=
a
x
component of
b
along
a
a
.
b
=
a
b
cos
θ
=
(
a
cos
θ
)
x
b
=
b
x
component of
a
along
b
a
.
b
=
a
b
cos
θ
a
.
b
=
a
b
cos
θ
=
a
x
(
b
cos
θ
)
=
a
x
component of
b
along
a
a
.
b
=
a
b
cos
θ
=
(
a
cos
θ
)
x
b
=
b
x
component of
a
along
b

The dot product of two vectors is equal to the arithmetic product of magnitude of one vector and component of second vector in the direction of first vector.

5: The cosine of the angle between two vectors can be obtained in terms of dot product as :

a
.
b
=
a
b
cosθ
a
.
b
=
a
b
cosθ

⇒
cosθ
=
a
.
b
a
b
⇒
cosθ
=
a
.
b
a
b

6: The condition of two perpendicular vectors in terms of dot product is given by :

a
.
b
=
a
b
cos
90
°
=
0
a
.
b
=
a
b
cos
90
°
=
0

7: Properties of dot product with respect to unit vectors along the axes of rectangular coordinate system are :

i
.
i
=
j
.
j
=
k
.
k
=
1
i
.
j
=
j
.
k
=
k
.
i
=
0
i
.
i
=
j
.
j
=
k
.
k
=
1
i
.
j
=
j
.
k
=
k
.
i
=
0

8: Dot product in component form is :

a
.
b
=
a
x
b
x
+
a
y
b
y
+
a
z
b
z
a
.
b
=
a
x
b
x
+
a
y
b
y
+
a
z
b
z