Vector Calculator
Perform comprehensive vector operations including addition, subtraction, dot product, cross product, and more
Vector Operations
Vector Addition (a + b)
Calculation Steps:
a + b = [2, 3] + [1, -2]
Result = [3.000, 1.000]
Example: Vector Addition
Physics Problem
Vector A (Force 1): [2, 3] N
Vector B (Force 2): [1, -2] N
Problem: Find the resultant force when both forces act on an object
Solution
Resultant Force = A + B = [2, 3] + [1, -2]
Resultant Force = [2 + 1, 3 + (-2)] = [3, 1] N
Magnitude = √(3² + 1²) = √10 ≈ 3.162 N
The resultant force is [3, 1] N with magnitude 3.162 N
Vector Operations
Addition
Component-wise addition
[a₁+b₁, a₂+b₂, a₃+b₃]
Dot Product
Scalar result, angle related
a₁b₁ + a₂b₂ + a₃b₃
Cross Product
Perpendicular vector
Right-hand rule applies
Vector Tips
Addition is commutative: a + b = b + a
Dot product: a · a = |a|²
Cross product is anti-commutative
Unit vectors have magnitude = 1
Understanding Vectors
What are Vectors?
Vectors are mathematical objects that have both magnitude (size) and direction. Unlike scalars, which only have magnitude, vectors can represent quantities like displacement, velocity, acceleration, and force in physics and engineering.
Vector Representation
- •Cartesian: [x, y] or [x, y, z]
- •Magnitude & Direction: |v| and θ (2D only)
- •Unit vectors: î, ĵ, k̂ notation
Key Formulas
Magnitude: |v| = √(x² + y² + z²)
Unit vector: û = v / |v|
Dot product: a · b = |a||b|cos(θ)
Cross product: |a × b| = |a||b|sin(θ)
Applications
- Physics: Forces, velocities, accelerations
- Engineering: Structural analysis, robotics
- Computer Graphics: 3D modeling, animations
- Navigation: GPS, flight paths
Vector Operations Reference
Geometric Operations
Addition (a + b)
Parallelogram law: Place tail of b at head of a
Result goes from tail of a to head of b
Dot Product (a · b)
Measures how much vectors point in same direction
Zero when vectors are perpendicular
Cross Product (a × b)
Creates vector perpendicular to both inputs
Magnitude equals parallelogram area
Properties & Rules
Commutative
a + b = b + a, a · b = b · a
Anti-commutative
a × b = -(b × a)
Distributive
a · (b + c) = a · b + a · c
Scalar Multiplication
k(a + b) = ka + kb