Vector Operations Tool

Vector operation complexity,
beautifully solved

A complete 2D & 3D vector calculator. Dot products, cross products, projections, angles — all with live visualization.

Open Calculator View Formulas
Vectors
A
Vector A
x
y
B
Vector B
x
y
Operation

Result will appear here

Visualization (2D projection)
Vector A Vector B
Formula Reference
Addition / Subtraction
A ± B = (ax±bx, ay±by, az±bz)

Resultant vector, component-wise
Dot Product
A · B = axbx + ayby + azbz

= |A||B| cos θ

Scalar result; zero if perpendicular
Cross Product (3D)
A × B = (aybz − azby,
          azbx − axbz,
          axby − aybx)

|A × B| = area of parallelogram
Magnitude
|A| = √(ax² + ay² + az²)

|A|² = A · A

Distance from origin to tip
Unit Vector
 = A / |A|

|Â| = 1 always

Direction without magnitude
Angle Between Vectors
θ = arccos( A·B / (|A||B|) )

Range: 0° to 180°

θ = 90° ⟺ A · B = 0
Projection of A onto B
proj_B A = (A·B / |B|²) × B

scalar proj = A·B / |B|

Component of A along B
Linear Combination
αA + βB

Any vector in the span of A, B
can be written this way