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Dot Product of Two Vectors

Last updated date: 13th Jul 2024
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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. The 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 the dot product of vectors, dot product definition, dot product formula, and dot product example in detail. Don't Miss Out: Get Your Free JEE Main Rank Predictor 2024 Instantly! 🚀

Dot Product Definition

The dot product of two different vectors that are non-zero  is denoted by a.b and is given by:

a.b = ab cos θ

wherein θ is the angle formed between a 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 a and b is denoted by:

a.b = |a||b| cos θ

Here, |a| and |b| are called the magnitudes of vectors 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)

Properties of Dot Product of Two Vectors 

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

Key Points to Remember

  • When two vectors are cross-products, the output is a vector that is orthogonal to the two provided vectors.

  • The right-hand thumb rule determines the direction of the cross product of two vectors, and the magnitude is determined by the area of the parallelogram generated by the original two vectors.

  • A zero vector is the cross-product of two linear vectors or parallel vectors.


Vector is a quantity that has both magnitude as well as direction. Few mathematical operations can be applied to vectors such as addition and multiplication. The multiplication of vectors can be done in two ways, i.e. dot product and cross product. The dot product of two vectors is the sum of the products of their corresponding components. It is the product of their magnitudes multiplied by the cosine of the angle between them. A vector's dot product with itself is the square of its magnitude.

FAQs on Dot Product of Two Vectors

1. What are the properties of Dot Product?

Following are the properties of dot product if a, b, and c are real vectors and r is a scalar:

  • Property 1: Commutative

  • Property 2: Distributive over vector addition – Vector product of two vectors always happens to be a vector

  • Property 3: Bilinear

  • Property 4: Scalar Multiplication

  • Property 5: Not associative 

  • Property 6: Orthogonal

  • Property 7: No cancelation

  • Property 8: Product Rule

2. What is a cross product of a two vector method?

The process of multiplying two vectors is called the cross product. A cross product is defined as the multiplication sign(x) of two vectors. It's a three-dimensional system with a binary vector function. The cross-product of two vectors is the third vector that is perpendicular to the two original vectors. Its magnitude is determined by the size of the parallelogram between them, and its direction is determined by the right-hand thumb rule. Because the outcome of the cross product of vectors is a vector quantity, the cross product of two vectors is also known as a vector product. When two vectors are cross-products, the output is a vector that is orthogonal to the two input vectors.

3. What is the significance of the dot product?

The concept of dot product says that two different vectors can be multiplied together to get the scalar quantity. It is used to get the product. It gives the products of two or even more vectors in two or more dimensions.