Dot Product Calculator

Vector A Components
Vector B Components

Result:

Calculate the Dot Product (scalar product) of two vectors with our comprehensive calculator. Understand vector relationships, find angles between vectors, and explore applications in physics and engineering.

What is the Dot Product?

The dot product (scalar product) of two vectors is a scalar quantity that measures how much one vector extends in the direction of another. It combines both magnitude and directional information.

Dot Product Formulas

Component Form

2D Vectors:

A⃗ · B⃗ = A₁B₁ + A₂B₂

3D Vectors:

A⃗ · B⃗ = A₁B₁ + A₂B₂ + A₃B₃

General:

A⃗ · B⃗ = Σ(AᵢBᵢ)

Geometric Form

Magnitude-Angle Formula:

A⃗ · B⃗ = |A⃗| |B⃗| cos θ

Where:

  • |A⃗| = magnitude of vector A
  • |B⃗| = magnitude of vector B
  • θ = angle between vectors

Step-by-Step Calculation

Example: A⃗ = (3, 4) and B⃗ = (2, -1)

Step 1: Identify components

A⃗ = (3, 4), so A₁ = 3, A₂ = 4

B⃗ = (2, -1), so B₁ = 2, B₂ = -1

Step 2: Apply dot product formula

A⃗ · B⃗ = A₁B₁ + A₂B₂

A⃗ · B⃗ = (3)(2) + (4)(-1)

A⃗ · B⃗ = 6 - 4 = 2

Step 3: Calculate magnitudes

|A⃗| = √(3² + 4²) = √25 = 5

|B⃗| = √(2² + (-1)²) = √5 ≈ 2.24

Step 4: Find angle

cos θ = (A⃗ · B⃗)/(|A⃗| |B⃗|) = 2/(5 × √5) = 2/(5√5)

θ = arccos(2/(5√5)) ≈ 73.9°

Properties of Dot Product

  • Commutative: A⃗ · B⃗ = B⃗ · A⃗
  • Distributive: A⃗ · (B⃗ + C⃗) = A⃗ · B⃗ + A⃗ · C⃗
  • Scalar multiplication: (kA⃗) · B⃗ = k(A⃗ · B⃗)
  • Self dot product: A⃗ · A⃗ = |A⃗|²
  • Zero vector: A⃗ · 0⃗ = 0

Geometric Interpretation

A⃗ · B⃗ > 0

Acute angle (θ < 90°)

Vectors point in similar directions

cos θ > 0
A⃗ · B⃗ = 0

Right angle (θ = 90°)

Vectors are perpendicular

cos θ = 0
A⃗ · B⃗ < 0

Obtuse angle (θ > 90°)

Vectors point in opposite directions

cos θ < 0

Applications of Dot Product

Physics Applications

Work calculation: W = F⃗ · d⃗

Power: P = F⃗ · v⃗

Flux: Φ = B⃗ · A⃗

Projection: Component of force

Engineering Applications

Computer graphics: Lighting calculations

Signal processing: Correlation

Machine learning: Similarity measures

Structural analysis: Force components

Vector Projection

The dot product is used to find the projection of one vector onto another:

Projection Formulas

Scalar projection of A⃗ onto B⃗:

proj_B⃗(A⃗) = (A⃗ · B⃗) / |B⃗|

Vector projection of A⃗ onto B⃗:

proj_B⃗(A⃗) = ((A⃗ · B⃗) / |B⃗|²) B⃗

Alternative form:

proj_B⃗(A⃗) = ((A⃗ · B⃗) / (B⃗ · B⃗)) B⃗

Special Cases

Unit Vectors

Standard unit vectors:

î = (1, 0, 0), ĵ = (0, 1, 0), k̂ = (0, 0, 1)

Properties:

î · î = ĵ · ĵ = k̂ · k̂ = 1

î · ĵ = ĵ · k̂ = k̂ · î = 0

Parallel Vectors

Same direction (θ = 0°):

A⃗ · B⃗ = |A⃗| |B⃗|

Opposite direction (θ = 180°):

A⃗ · B⃗ = -|A⃗| |B⃗|

Practice Problems

2D Vectors

A⃗ = (1, 2), B⃗ = (3, -1)

Answer: 1

1×3 + 2×(-1) = 1
3D Vectors

A⃗ = (2, 1, 3), B⃗ = (1, -1, 2)

Answer: 7

2×1 + 1×(-1) + 3×2 = 7
Perpendicular Test

A⃗ = (1, 2), B⃗ = (2, -1)

Answer: 0

Perpendicular vectors

Computational Tips

  • Order of operations: Multiply corresponding components first, then sum
  • Sign checking: Pay attention to negative components
  • Magnitude calculation: Use Pythagorean theorem in multiple dimensions
  • Angle calculation: Use inverse cosine for angle between vectors
  • Verification: Check if result makes geometric sense

Calculator Categories

Explore our comprehensive collection of calculation tools organized by category. Find exactly what you need for math, science, finance, health, and more.

26

Categories
100+ Calculators
Instant Results
Search Calculators

All Categories

Choose from our specialized calculator categories

Algebra

Comprehensive algebra calculators for equations, roots, exponents, logarithms, and more

22 calculators
Explore Algebra
Area Converters

Convert between different area units including square feet, square meters, and acres

2 calculators
Explore Area Converters
Chemistry

<p>Chemistry can be a complex subject, but it doesn't have to be overwhelming! With our powerful ch…

1 calculator
Explore Chemistry
Construction

Construction calculators.

1 calculator
Explore Construction
Conversions

In today's interconnected world, converting units and measurements is a common task. But who has ti…

23 calculators
Explore Conversions
Cooking Converters

Convert between cooking measurement units including teaspoons, tablespoons, cups, and ounces

5 calculators
Explore Cooking Converters
Data Storage Converters

Convert between different data storage units including bytes, kilobytes, megabytes, and gigabytes

2 calculators
Explore Data Storage Converters
Energy Converters

Convert between different energy units including joules, calories, kWh, and BTU

1 calculator
Explore Energy Converters
Everyday Life

<p>In our busy daily lives, we often encounter situations that require quick calculations. Whether …

6 calculators
Explore Everyday Life
Finance

<p>Our finance calculators help you make smart choices about money. Whether you're saving up for so…

3 calculators
Explore Finance
Fractions

Comprehensive fraction calculators for all fraction operations

16 calculators
Explore Fractions
Health

<p>Keeping track of your health can be a challenge, but it doesn't have to be! With our amazing hea…

3 calculators
Explore Health
Length Converters

Convert between different length and distance units including meters, feet, inches, and miles

5 calculators
Explore Length Converters
Maths

Math can seem like a tough subject, but it doesn't have to be! With our awesome math calculator, yo…

87 calculators
Explore Maths
Percentage

Comprehensive percentage calculators for discounts, taxes, tips, and voting calculations

4 calculators
Explore Percentage
Physics Converters

Convert between physics units including acceleration, force, density, and angles

4 calculators
Explore Physics Converters
Power Converters

Convert between different power units including watts, horsepower, and BTU/hr

1 calculator
Explore Power Converters
Pressure Converters

Convert between different pressure units including PSI, bar, and atmospheres

2 calculators
Explore Pressure Converters
Speed Converters

Convert between different speed units including mph, kph, m/s, and knots

1 calculator
Explore Speed Converters
Sports

p>In the world of sports, even the slightest edge can make a big difference. Whether you're a profe…

1 calculator
Explore Sports
Statistics

Statistical calculators for data analysis, probability, and descriptive statistics

16 calculators
Explore Statistics
Temperature Converters

Convert between Celsius, Fahrenheit, and Kelvin temperature scales

3 calculators
Explore Temperature Converters
Time Converters

Convert between different time units including seconds, minutes, hours, days, and years

5 calculators
Explore Time Converters
Time and Date

<p>Keeping track of dates, times, and schedules can be a daunting task. Whether you're planning a p…

3 calculators
Explore Time and Date
Volume Converters

Convert between different volume and capacity units including liters, gallons, cups, and milliliters

8 calculators
Explore Volume Converters
Weight Converters

Convert between different weight and mass units including pounds, kilograms, ounces, and grams

4 calculators
Explore Weight Converters