Dot Product Calculator
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 > 0A⃗ · B⃗ = 0
Right angle (θ = 90°)
Vectors are perpendicular
cos θ = 0A⃗ · B⃗ < 0
Obtuse angle (θ > 90°)
Vectors point in opposite directions
cos θ < 0Applications 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) = 13D Vectors
A⃗ = (2, 1, 3), B⃗ = (1, -1, 2)
Answer: 7
2×1 + 1×(-1) + 3×2 = 7Perpendicular Test
A⃗ = (1, 2), B⃗ = (2, -1)
Answer: 0
Perpendicular vectorsComputational 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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