Dot Product Calculator
Calculate the scalar product of two vectors and find the angle between them
Vector Input
Dimension Mode
Vector A Components
Vector A: [4, 5, -3]
Magnitude |A|: 7.0711
Vector B Components
Vector B: [1, -2, -2]
Magnitude |B|: 3.0000
Calculation Results
Vector Relationship: Perpendicular (Orthogonal)
The vectors are perpendicular (orthogonal) to each other.
Step-by-Step Solution
Example Calculation
Perpendicular Vectors Example
Problem: Find the dot product of vectors a = [1, 1] and b = [1, -1]
Vector a: [1, 1]
Vector b: [1, -1]
Solution
Step 1: Apply dot product formula: a·b = a₁b₁ + a₂b₂
Step 2: Substitute values: a·b = (1)(1) + (1)(-1)
Step 3: Calculate: a·b = 1 + (-1) = 0
Step 4: Since the dot product is 0, the vectors are perpendicular
Result: Dot Product = 0, Angle = 90°
Dot Product Formulas
Algebraic Form
2D: a·b = a₁b₁ + a₂b₂
3D: a·b = a₁b₁ + a₂b₂ + a₃b₃
Geometric Form
a·b = |a| x |b| x cos(α)
Where α is the angle between vectors
Angle Formula
cos(α) = (a·b) / (|a| x |b|)
α = arccos[(a·b) / (|a| x |b|)]
Quick Reference
Dot product = 0 → Vectors are perpendicular
Angle = 0° → Vectors are parallel (same direction)
Angle = 180° → Vectors are parallel (opposite direction)
Dot product > 0 → Acute angle (< 90°)
Dot product < 0 → Obtuse angle (> 90°)
Understanding the Dot Product
What is the Dot Product?
The dot product (also called scalar product) is a mathematical operation that takes two vectors and returns a single number (scalar). It measures how much one vector extends in the direction of another vector.
Geometric Interpretation
Geometrically, the dot product represents the product of the magnitudes of two vectors and the cosine of the angle between them. It can be thought of as the projection of one vector onto another, multiplied by the magnitude of the second vector.
Properties
- •Commutative: a·b = b·a
- •Distributive: a·(b + c) = a·b + a·c
- •Scalar multiplication: (ka)·b = k(a·b)
- •Self dot product: a·a = |a|²
Real-World Applications
- •Physics: Work = Force · Displacement
- •Computer Graphics: Lighting calculations
- •Machine Learning: Similarity measures
- •Signal Processing: Correlation analysis
Special Cases
Perpendicular vectors: When a·b = 0, the vectors are orthogonal (90° angle)
Parallel vectors: When |cos(α)| = 1, vectors are parallel or anti-parallel
Unit vectors: For unit vectors, a·b = cos(α) directly gives the cosine of the angle
Calculation Methods
Component Method
Most common approach using vector components:
- Multiply corresponding components
- Sum all the products
- The result is the dot product
Magnitude-Angle Method
Using magnitudes and angle between vectors:
- Calculate magnitude of each vector
- Determine angle between vectors
- Apply: a·b = |a| × |b| × cos(α)