Vector Calculator - Vector Operations & Analysis

Vector Addition & Subtraction

Vector Dimension:

Vector A:

Vector B:

Vector Addition Rules

Vector addition is commutative (A⃗ + B⃗ = B⃗ + A⃗) and associative ((A⃗ + B⃗) + C⃗ = A⃗ + (B⃗ + C⃗)). Geometrically, it follows the parallelogram rule.

Dot Product & Cross Product

Vector A:

Vector B:

Dot Product: A⃗ · B⃗ = |A⃗||B⃗|cos(θ) = AxBx + AyBy + AzBz Cross Product: A⃗ × B⃗ = |A⃗||B⃗|sin(θ)n̂

Product Properties

Dot Product: Results in a scalar. Zero when vectors are perpendicular.
Cross Product: Results in a vector perpendicular to both input vectors (3D only).

Vector Properties & Analysis

Vector:

Vector from Points

Point A

Point B

|V⃗| = √(x² + y² + z²) Unit Vector: û = V⃗/|V⃗| Direction Angles: cos(α) = Vx/|V⃗|, cos(β) = Vy/|V⃗|, cos(γ) = Vz/|V⃗|

Vector Projections

Vector A⃗:

Vector B⃗:

Scalar Projection: comp_B⃗(A⃗) = A⃗ · B̂ = (A⃗ · B⃗)/|B⃗| Vector Projection: proj_B⃗(A⃗) = ((A⃗ · B⃗)/|B⃗|²) × B⃗ Vector Rejection: A⃗ - proj_B⃗(A⃗)

Projection Applications

Vector projections are used in physics for force components, computer graphics for lighting calculations, and data analysis for principal component analysis.

Lines and Planes

Point on Line/Plane

P₀:

Direction/Normal Vector

V⃗:

Distance Calculations

Distance Type:

Line: r⃗ = r⃗₀ + td⃗ (parametric) Plane: n⃗ · (r⃗ - r⃗₀) = 0 or ax + by + cz = d Distance point to plane: |n⃗ · (P⃗ - P⃗₀)|/|n⃗|

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