Admissions 2025-26

☰

Go Cosmo 2024

Dot Product of Two Vectors

Introduction

In mathematics, vectors are important tools in geometry, physics, engineering, and computer science. They describe quantities that have both size and direction. The dot product of two vectors, also called the scalar product, is one method of combining vectors to determine how much one vector points in the direction of another. This blog will explain everything you need to know about the dot product, including its formula, real-world uses, and practice problems.

Table of Contents

What Is a Vector?
Understanding the Dot Product
Dot Product Formula
Geometric Meaning of Dot Product
Algebraic Method to Calculate Dot Product
Step-by-Step Example of Dot Product
Dot Product in Coordinate Geometry
Dot Product of Unit Vectors
Dot Product of Two Vectors in 3D
Properties of Dot Product
Difference Between Dot Product and Cross Product
Real-Life Applications of Dot Product
Common Mistakes While Using Dot Product
Conclusion
FAQs On Dot Product of Two Vectors

What Is a Vector?

A vector is a mathematical quantity with both magnitude (length) and direction. Vectors are usually represented as arrows or in coordinate form, such as:

2D vector: A = 3i + 4j
3D vector: B = 2i + 5j + 6k

Here, i, j, and k are unit vectors in the directions of the x-, y-, and z-axes, respectively.

Understanding the Dot Product

The dot product is a method of multiplying two vectors that results in a scalar (a number, not a vector). It helps us find:

The angle between two vectors
The projection of one vector onto another
Whether vectors are perpendicular (dot product = 0)

The value tells you how "aligned" the vectors are.

Dot Product Formula

Two formulas depend on what information you have:

Geometric Formula:

A · B = |A||B|cosθ
Where:

A and B are vectors
|A| and |B| are magnitudes
θ is the angle between them

This tells us how much of one vector acts in the direction of another.

Algebraic Formula (Product of Vectors Formula):

If A = a₁i + a₂j + a₃k and B = b₁i + b₂j + b₃k,
then:
A · B = a₁b₁ + a₂b₂ + a₃b₃

Geometric Meaning of Dot Product

The dot product relates directly to the angle between vectors. Important cases:

θ = 0°: Vectors are in the same direction → A · B = |A||B|
θ = 90°: Vectors are perpendicular → A · B = 0
θ = 180°: Vectors point in opposite directions → A · B = -|A||B|

Thus, the dot product tells us not just the amount but the nature of the interaction between the vectors.

Algebraic Method to Calculate Dot Product

Let’s take an example:

A = 2i + 3j + 4k
B = 1i + 0j + 5k

Now use the algebraic formula:
A · B = (2)(1) + (3)(0) + (4)(5) = 2 + 0 + 20 = 22

So the dot product = 22

Step-by-Step Example of Dot Product

Problem:

Find the dot product of vectors
A = 6i - 2j + 3k
B = 4i + j - k

Solution:
A · B = (6)(4) + (-2)(1) + (3)(-1)
= 24 - 2 - 3 = 19

Answer: 19

Dot Product in Coordinate Geometry

In coordinate geometry, the dot product helps in checking the angle between two vectors:

If A = (x₁, y₁) and B = (x₂, y₂), then:
A · B = x₁x₂ + y₁y₂

If this equals zero, the vectors are perpendicular.

Example:
A = (1, 2), B = (2, -1)
Dot product = (1)(2) + (2)(-1) = 2 - 2 = 0
→ A and B are perpendicular.

Dot Product of Unit Vectors

Unit vectors are vectors of length 1.
The dot products of unit vectors are:

i · i = 1, j · j = 1, k · k = 1
i · j = 0, i · k = 0, j · k = 0

This shows that i, j, and k are all perpendicular to each other.

Dot Product of Two Vectors in 3D

Let’s revisit:
A = 2i + 4j + 5k
B = 3i + 2j + k

Dot Product = (2)(3) + (4)(2) + (5)(1) = 6 + 8 + 5 = 19

Properties of Dot Product

Commutative: A · B = B · A
Distributive: A · (B + C) = A · B + A · C
Scalar multiplication: (kA) · B = k(A · B)
Zero vector rule: A · 0 = 0
Orthogonal test: A · B = 0 ⇒ vectors are perpendicular
Self-product: A · A = |A|²

Difference Between Dot Product and Cross Product

Feature	Dot Product	Cross Product
Output Type	Scalar	Vector
Symbol Used	·	×
Angle Function	Cosθ	Sinθ
Geometric Meaning	Projection/Alignment	Perpendicular vector
Applications	Work, angle, similarity	Torque, rotation direction

Real-Life Applications of Dot Product

Physics:
Work = Force · Displacement
If the force and motion are perpendicular, no work is done.
Robotics:
To determine alignment between joint directions.
3D Graphics:
Used in shading, lighting, and object orientation.
Data Science:
Cosine similarity is calculated using the dot product to compare text or features.
Aviation & Navigation:
Calculate angles between directional paths.

Common Mistakes While Using Dot Product

Confusing it with the cross product: Remember, the dot product gives a number, not a vector.
Forgetting zero terms: Any zero component still counts and must be added.
Using degrees: Radians (or vice versa) while applying cos(θ).
Skipping brackets: Always write (a)(b) when multiplying components.

Conclusion

The dot product of two vectors is a straightforward but very useful tool in math and science. It helps us understand angles, projections, similarity, and even physical quantities like work and energy. Whether you are solving simple vector equations or dealing with advanced applications in robotics, data science, or physics, the dot product is a key concept.

By mastering the dot product formula and its meanings, you gain a deeper understanding of how vectors interact in the real world. You can calculate that interaction quickly and accurately using basic algebra.

Related Topics

Addition - Add with Confidence! Learn how to add numbers quickly and accurately with simple techniques and everyday examples. Perfect your addition skills today!

Subtraction - Subtract Like a Star! Master subtraction using easy steps, real-life examples, and practice questions. Make subtraction simple and fun!

Multiplication - Multiply Smarter, Not Harder! Discover easy tricks and strategies to boost your multiplication skills. Build speed and accuracy in no time!

FAQs On Dot Product of Two Vectors

What is a dot product of two vectors?

The dot product is a way to multiply two vectors to get a scalar. It tells how much one vector points in the direction of the other. If the result is 0, the vectors are perpendicular.