Dot Product of Two Vectors – Properties and Examples
Vectors can be multiplied in two different ways, namely, scalar product or dot product in which the result is a scalar, and vector product or cross product in which the result is a vector. Dot product of two vectors means the scalar product of the two given vectors. It is a scalar number that is obtained by performing a specific operation on the different vector components. The dot product is applicable only for the pairs of vectors that have the same number of dimensions. The symbol that is used for the dot product is a heavy dot. This dot product is widely used in mathematics and Physics. In this article, we would be discussing about the dot product of vectors, dot product definition, dot product formula and dot product example in detail.
Dot Product Definition
The dot product of two different vectors and that are nonzero and denoted by a.bis given by:
ab = ab cos θ
wherein is the angle formed between and b, and,
0 ≤ θ ≤ π
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If a = 0 or b = 0, θ will not be defined, and in this case,
a.b= 0
Dot Product Formula
You can define the dot product of two vectors in two different methods: geometrically and algebraically.
Dot Product Geometry Definition
The geometric meaning of dot product says that the dot product between two given vectors and bis denoted by:
a⋅b = a ∣∣ b ∣ cos
Here, a and b are called as the magnitudes of vector a and b and θ is the angle between the vectors a and b.
If the two vectors are orthogonal, that is, the angle between them is 90, then a.b = 0 since cos 90 = 0.
If the two vectors are parallel to each other, then a.b =ab since cos 0 = 1.
Dot Product Algebra Definition
The dot product algebra says that the dot product of the given two products – a = (a_{1}, a_{2}, a_{3}) and b= (b_{1}, b_{2}, b_{3}) is given by:
a.b= (a_{1}b_{1} + a_{2}b_{2} + a_{3}b_{3})
Dot Product of Two Vectors Properties
Given below are the properties of vectors:

Commutative Property
a .b = b.a
a.b =a bcos θ
a.b =bacos θ

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

Bilinear Property
a.(rb + c) = r.(a.b) + (a.c)

Scalar Multiplication Property
(xa) . (yb) = xy (a.b)

NonAssociative Property
Since the dot product between a scalar and a vector is not allowed

Orthogonal Property
Two vectors are orthogonal only when a.b = 0
Dot Product of Vector – Valued Functions
The dot product of vectorvalued functions, that are r(t) and u(t) each gives you a vector at each particular time t, and hence the function r(t)⋅u(t) is said to be a scalar function.
Solved Examples
Example 1:
Find the dot product of a= (1, 2, 3) and b= (4, −5, 6) . What kind of angle the vectors would form?
Solution:
Using the formula of the dot products,
a.b = (a_{1}b_{1} + a_{2}b_{2} + a_{3}b_{3})
You can calculate the dot product to be
= 1(4) + 2(5) + 3(6)
= 4 – 10 + 18
= 12
Since a.b is a positive number you can infer that the vectors would form an acute angle.
Example 2:
Two vectors A and B are given by:
A = 2i – 3j + 7k and B= 4i + 2j 4k
Find the dot product of the given two vectors.
Solution:
A.B = (2i – 3j +7k) . (4i + 2j – 4k)
= 2 (4) + (3)2 + 7 (4)
= 8 – 6 – 28
= 42