Vector Calculator - Vector Operations

This physics calculator helps you calculate vector operations including addition, subtraction, magnitude, dot product, and cross product. Vectors are quantities with both magnitude and direction.

Vector Definition

• Vector: Quantity with magnitude and direction
• Magnitude: |v| = √(x² + y² + z²)
• Unit Vector: v̂ = v/|v| (magnitude = 1)
• Components: v = (x, y, z)

Vector Operations

• Addition: v₁ + v₂ = (x₁+x₂, y₁+y₂, z₁+z₂)
• Subtraction: v₁ - v₂ = (x₁-x₂, y₁-y₂, z₁-z₂)
• Scalar Multiplication: av = (ax, ay, az)
• Dot Product: v₁·v₂ = x₁x₂ + y₁y₂ + z₁z₂
• Cross Product: v₁×v₂ = (y₁z₂-z₁y₂, z₁x₂-x₁z₂, x₁y₂-y₁x₂)

Applications

• Mechanics: Force, velocity, acceleration
• Electromagnetism: Electric and magnetic fields
• Geometry: Position, displacement, direction
• Computer Graphics: 3D modeling and animation
• Navigation: GPS and compass directions
• Engineering: Structural analysis and design

Important Notes

• Vectors have magnitude and direction
• Dot product gives scalar result
• Cross product gives vector result
• Cross product is perpendicular to both vectors
• Unit vectors have magnitude 1

Vector Properties

• Commutative: v₁ + v₂ = v₂ + v₁
• Associative: (v₁ + v₂) + v₃ = v₁ + (v₂ + v₃)
• Distributive: a(v₁ + v₂) = av₁ + av₂
• Dot Product: v₁·v₂ = |v₁||v₂|cosθ
• Cross Product: |v₁×v₂| = |v₁||v₂|sinθ

Vector Examples

• Position Vector: r = (x, y, z)
• Velocity Vector: v = (vx, vy, vz)
• Force Vector: F = (Fx, Fy, Fz)
• Electric Field: E = (Ex, Ey, Ez)
• Magnetic Field: B = (Bx, By, Bz)

Vector Coordinate Systems

• Cartesian: (x, y, z)
• Cylindrical: (r, θ, z)
• Spherical: (r, θ, φ)
• 2D: (x, y) or (r, θ)

Vector Relationships

• Perpendicular: v₁·v₂ = 0
• Parallel: v₁×v₂ = 0
• Orthogonal: v₁·v₂ = 0 and v₁×v₂ ≠ 0
• Collinear: v₁×v₂ = 0
• Angle: cosθ = (v₁·v₂)/(|v₁||v₂|)

Vector Calculus

• Gradient: ∇f = (∂f/∂x, ∂f/∂y, ∂f/∂z)
• Divergence: ∇·F = ∂Fx/∂x + ∂Fy/∂y + ∂Fz/∂z
• Curl: ∇×F = (∂Fz/∂y-∂Fy/∂z, ∂Fx/∂z-∂Fz/∂x, ∂Fy/∂x-∂Fx/∂y)
• Laplacian: ∇²f = ∂²f/∂x² + ∂²f/∂y² + ∂²f/∂z²