Dot Product of Two Vectors – Properties and Examples

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 non-zero and denoted by a.bis given by:

ab = ab cos θ

wherein is the angle formed between and b, and,

0 ≤ θ ≤ π



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 =|a||b| since cos 0 = 1.

Dot Product Algebra Definition

The dot product algebra says that the dot product of the given two products – a = (a1, a2, a3) and b= (b1, b2, b3) is given by:

a.b= (a1b1 + a2b2 + a3b3)

Dot Product of Two Vectors Properties

Given below are the properties of vectors:

  1. Commutative Property

a .b = b.a

a.b =|a| b|cos θ

a.b =|b||a|cos θ

  1. Distributive Property

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

  1. Bilinear Property

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

  1. Scalar Multiplication Property

(xa) . (yb) = xy (a.b)

  1. Non-Associative Property

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

  1. Orthogonal Property

Two vectors are orthogonal only when a.b = 0

Dot Product of Vector – Valued Functions

The dot product of vector-valued 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?


Using the formula of the dot products,

a.b = (a1b1 + a2b2 + a3b3)

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.


A.B = (2i – 3j +7k) . (-4i + 2j – 4k)

= 2 (-4) + (-3)2 + 7 (-4)

= -8 – 6 – 28

= -42

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