Vector Dot Product (Scalar Product)
⭕Definition and Formula
a · b = |a| |b| cos(θ)
where θ is the angle between vectors a and b
Component Form
a · b = a₁b₁ + a₂b₂ + a₃b₃ + ... + aₙbₙ
2D Vectors
a · b = a₁b₁ + a₂b₂
Two components
3D Vectors
a · b = a₁b₁ + a₂b₂ + a₃b₃
Three components
Result
Always a scalar
Not a vector
🎯Geometric Interpretation
Projection Interpretation
a · b = |a| × (projection of b onto a)
Meaning: Magnitude of a times the component of b in direction of a
The dot product measures how much two vectors "point in the same direction".
Angle Between Vectors
cos(θ) = (a · b) / (|a| |b|)
Finding angle:
θ = arccos((a · b) / (|a| |b|))
Most common use of dot product
⚡Properties of Dot Product
Commutative
a · b = b · a
Order doesn't matter
Unlike cross product
Distributive
a · (b + c) = a · b + a · c
Distributes over addition
Very useful property
Scalar Multiplication
(ca) · b = c(a · b)
Scalar can be factored out
Also: a · (cb) = c(a · b)
🔄Special Cases
Perpendicular Vectors
a · b = 0
When: θ = 90°
Orthogonal vectors
cos(90°) = 0
Parallel Vectors
a · b = ±|a| |b|
When: θ = 0° or 180°
+ for same direction
− for opposite direction
Dot with Itself
a · a = |a|²
Result: Square of magnitude
Always positive (unless zero vector)
θ = 0° with itself
📐Common Applications
Finding Angles
Formula:
cos(θ) = (a · b) / (|a| |b|)
- • Used in 3D graphics
- • Lighting calculations
- • Camera angles
Testing Orthogonality
Test:
If a · b = 0, then perpendicular
- • Collision detection
- • Surface normals
- • Coordinate systems
📊Physics: Work and Energy
W = F · d
Work = Force · Displacement
θ = 0°
Maximum work
Force in direction of motion
θ = 90°
Zero work
Force perpendicular to motion
θ = 180°
Negative work
Force opposes motion
🔬 Real-World Applications
- • Computer Graphics: Lighting and shading calculations
- • Physics: Work, energy, and power calculations
- • Machine Learning: Similarity measures, cosine similarity
- • Game Development: Collision detection, AI behavior
- • Signal Processing: Correlation and filtering
🧮 Problem-Solving Tips
- • Use component form for calculation
- • Use geometric form for understanding
- • Check if vectors are perpendicular (dot = 0)
- • Remember: result is always a scalar
- • Normalize vectors for cosine similarity
- • Dot product is commutative