Vector Dot Product Calculator

Calculator and formulas for computing the dot product (scalar product) of two vectors

Vector Dot Product Calculator

Vector Dot Product (Scalar Product)

Calculates the dot product v₁ · v₂ of two vectors as a real number (scalar): x₁×x₂ + y₁×y₂ + z₁×z₂

Result is a Scalar (Real Number)

Unlike other vector operations, the dot product produces no vector, but a real number.

Select Vector Dimension
First Vector (v₁)
Second Vector (v₂)
Dot Product Result
Dot Product (v₁ · v₂):
Components are multiplied pairwise and added: v₁ · v₂ = x₁×x₂ + y₁×y₂ + ...

Dot Product Info

Dot Product Properties

Result: Real number (not a vector!)

Scalar Commutative v₁ · v₂

Commutative: v₁ · v₂ = v₂ · v₁
Orthogonal: v₁ ⊥ v₂ ⟺ v₁ · v₂ = 0

Examples
[1, 2, 3] · [4, 5, 6] = 32
[1, 0] · [0, 1] = 0 (orthogonal)
[2, 3] · [2, 3] = 13 (magnitude²)

Formulas for the Dot Product

Basic Formula
\[\vec{a} \cdot \vec{b} = \sum_{i=1}^{n} a_i \cdot b_i\]

General summation formula

2D Dot Product
\[\left[\matrix{a\\b}\right] \cdot \left[\matrix{c\\d}\right] = ac + bd\]

Two-dimensional vectors

3D Dot Product
\[\left[\matrix{a\\b\\c}\right] \cdot \left[\matrix{x\\y\\z}\right] = ax + by + cz\]

Three-dimensional vectors

Geometric Form
\[\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|\cos\theta\]

With angle between vectors

Calculation Examples for Dot Product

Example 1: 3D Dot Product
[1, 2, 3] · [4, 5, 6]
\[\begin{aligned} \vec{v_1} \cdot \vec{v_2} &= 1 \cdot 4 + 2 \cdot 5 + 3 \cdot 6 \\ &= 4 + 10 + 18 = 32 \end{aligned}\]

Result: 32 (Scalar)

Example 2: 2D Dot Product
[1, 2] · [4, 5]
\[\begin{aligned} \vec{u} \cdot \vec{w} &= 1 \cdot 4 + 2 \cdot 5 \\ &= 4 + 10 = 14 \end{aligned}\]

Result: 14 (Scalar)

Step-by-Step Calculation
1. Multiply: a₁×b₁
Corresponding components
2. Add: Σ(aᵢ×bᵢ)
Sum all products
3. Result: Scalar
Real number, not vector

The dot product always produces a real number

Geometric Interpretations
v · v = |v|²

• Dot product with itself

• Equals magnitude squared

• Always positive

v₁ · v₂ = 0

• Vectors orthogonal

• Perpendicular to each other

• θ = 90°

v₁ · v₂ > 0

• Acute angle

• θ < 90°

• Similar directions

v₁ · v₂ < 0

• Obtuse angle

• θ > 90°

• Opposite tendencies

Applications of the Dot Product

The dot product is fundamental in many areas of mathematics, physics and computer science:

Computer Graphics & 3D
  • Lighting calculations (Diffuse/Specular)
  • Determine angles between vectors
  • Backface culling and visibility
  • Calculate vector projections
Physics & Mechanics
  • Work: W = F · s (Force × Distance)
  • Orthogonality tests
  • Power and energy calculations
  • Angles between force vectors
Machine Learning
  • Cosine-similarity calculations
  • Neural networks: weighted sums
  • Feature comparisons and similarity
  • Gradient calculations
Mathematics & Statistics
  • Correlation and covariance
  • Linear algebra: orthogonal bases
  • Projections and approximations
  • Fourier transformations

Dot Product: From Vectors to Scalars

The dot product is a fundamental operation in linear algebra that assigns a real number to two vectors. This seemingly simple operation - multiplying corresponding components and adding them - carries profound geometric meaning. It measures both the similarity of vector directions and their relative lengths. From orthogonality tests through angle calculations to modern data science, the dot product remains an indispensable tool for quantitative analysis of vector relationships.

Summary

The dot product combines computational simplicity with geometric depth. The intuitive calculation - multiply and add - reveals fundamental relationships between vectors: orthogonality at zero, similarity at positive values, opposition at negative values. From 3D graphics through physical simulations to machine learning, the dot product enables precise quantification of vector relationships. It demonstrates how elegant mathematical operations make complex spatial and abstract relationships measurable and usable.

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Vector Functions

Addition • Subtraction • Multiplication • Scalar Multiplication • Division • Scalar Division • Dot Product • Cross Product • Interpolation • Distance • Distance Squaret • Normalization • Reflection • Magnitude • Squared-Magnitude • Triple-Product