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

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