Dot Product Calculator

Calculate dot product, angle between vectors, magnitudes, and projections for 2D and 3D vectors.

Vector Input

Vector 1 (v₁)

Vector 2 (v₂)

Dot Product Formula

2D: v₁ · v₂ = x₁×x₂ + y₁×y₂

3D: v₁ · v₂ = x₁×x₂ + y₁×y₂ + z₁×z₂

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About Dot Product Calculator

The dot product (also called scalar product or inner product) is a fundamental operation in vector mathematics. This calculator helps you compute the dot product of two vectors in both 2D and 3D space, along with related properties like magnitude, angle, and projection.

What is a Dot Product?

The dot product of two vectors is a scalar value that represents the product of their magnitudes and the cosine of the angle between them. It's calculated by multiplying corresponding components and summing the results.

Dot Product Formulas

  • 2D Vectors: v₁ · v₂ = x₁×x₂ + y₁×y₂
  • 3D Vectors: v₁ · v₂ = x₁×x₂ + y₁×y₂ + z₁×z₂
  • Angle Formula: cos(θ) = (v₁ · v₂) / (|v₁| × |v₂|)
  • Projection Formula: proj = (v₁ · v₂) / |v₂|

Key Properties

  • Commutative: v₁ · v₂ = v₂ · v₁
  • Distributive: v₁ · (v₂ + v₃) = v₁ · v₂ + v₁ · v₃
  • Scalar Multiplication: (cv₁) · v₂ = c(v₁ · v₂)
  • Orthogonality: If v₁ · v₂ = 0, vectors are perpendicular
  • Parallel Vectors: If angle is 0° or 180°, vectors are parallel

Applications

  • Physics: Calculate work done by a force (W = F · d)
  • Computer Graphics: Lighting calculations and surface normals
  • Machine Learning: Similarity measures and neural networks
  • Engineering: Stress analysis and structural mechanics
  • Geometry: Finding angles and projections between vectors

How to Use This Calculator

  1. Select whether you're working with 2D or 3D vectors
  2. Enter the x, y (and z for 3D) components for both vectors
  3. Click "Calculate" to see the dot product and all related properties
  4. View the step-by-step solution to understand the calculation process
  5. Check vector relationships (orthogonal, parallel, or neither)

Understanding the Results

  • Dot Product: The scalar result of the operation
  • Magnitudes: The length of each vector
  • Angle: The angle between the two vectors in degrees and radians
  • Projection: How much of v₁ points in the direction of v₂
  • Relationship: Whether vectors are orthogonal, parallel, or at an angle

Common Use Cases

Checking Perpendicularity: Two vectors are perpendicular if their dot product equals zero. This is useful in geometry and physics problems.

Finding Angles: Use the dot product to find the angle between two vectors, essential in navigation, robotics, and computer graphics.

Calculating Projections: Project one vector onto another to find components in specific directions, useful in physics and engineering.