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 algebraic 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
9: The dot product does not yield to cancellation. For example, if a.b = a.c, then we can not conclude that b = c. Rearranging, we have :
a
.
b

a
.
c
=
0
a
.
(
b

c
)
=
0
a
.
b

a
.
c
=
0
a
.
(
b

c
)
=
0
This means that a and (b  c) are perpendicular to each other. In turn, this implies that (b  c) is not equal to zero (null vector). Hence, b is not equal to c as we would get after cancellation.
We can understand this difference with respect to cancellation more explicitly by working through the problem given here :
Problem : Verify vector equality B = C, if A.B = A.C.
Solution : The given equality of dot products is :
A
.
B
=
A
.
C
A
.
B
=
A
.
C
We should understand that dot product is not a simple algebraic product of two numbers (read magnitudes). The angle between two vectors plays a role in determining the magnitude of the dot product. Hence, it is entirely possible that vectors B and C are different yet their dot products with common vector A are equal. Let
θ
1
θ
1
and
θ
2
θ
2
be the angles for first and second pairs of dot products. Then,
A
.
B
=
A
.
C
A
.
B
=
A
.
C
AB cos
θ
1
=
AC cos
θ
2
AB cos
θ
1
=
AC cos
θ
2
If
θ
1
=
θ
2
θ
1
=
θ
2
, then
B
=
C
B = C
. However, if
θ
1
≠
θ
2
θ
1
≠
θ
2
, then
B
≠
C
B ≠ C
.