2D Vector Calculator
Add, subtract, and compute dot product, cross product magnitude, unit vector, and angle between two 2D vectors.
Shows component-by-component working.
A vector has both magnitude (size) and direction, unlike a scalar which has only magnitude. Vectors are fundamental to physics, engineering, computer graphics, and machine learning.
Notation: A 2D vector is written as (x, y) or as a column matrix. A 3D vector is (x, y, z).
Basic vector operations:
Addition: (a, b) + (c, d) = (a+c, b+d) Example: (3, 2) + (1, 4) = (4, 6)
Subtraction: (a, b) − (c, d) = (a−c, b−d)
Scalar multiplication: k(a, b) = (ka, kb) Example: 3 × (2, −1) = (6, −3)
Magnitude (length): |v| = √(x² + y² + z²) Example: |(3, 4)| = √(9 + 16) = √25 = 5
Unit vector: v̂ = v / |v| (vector divided by its magnitude, giving length = 1) Example: (3, 4) / 5 = (0.6, 0.8)
Dot product: a·b = axbx + ayby + azbz Example: (2, 3) · (4, 1) = 8 + 3 = 11 Dot product = |a||b|cos(θ), so θ = arccos(a·b / (|a||b|))
Cross product (3D only): produces a vector perpendicular to both inputs (a × b)x = aybz − azby (a × b)y = azbx − axbz (a × b)z = axby − aybx
Worked example — finding angle between vectors: v1 = (1, 0), v2 = (1, 1) Dot product = 1, |v1| = 1, |v2| = √2 cos(θ) = 1 / √2 → θ = 45°
Applications:
- Physics: force, velocity, acceleration, electric fields
- Computer graphics: surface normals, lighting calculations, camera orientation
- Navigation: combining wind speed + aircraft speed to get ground speed
- Machine learning: word embeddings, similarity calculations
How we build and check this calculator
This calculator runs entirely in your browser, so the numbers you enter stay on your device. The math behind it is written by hand and tested against worked examples and standard references before the page goes live.
SuperGlobalCalculator is independently built and maintained. See how we build and verify our calculators.