How to Use the Vector Calculator

Select 2D or 3D mode, choose an operation type, enter your vector components, and click Calculate. Results include the computed vector and related properties.

Calculator Modes

Vector Arithmetic

Perform basic operations on vectors:

Addition (A + B): Add corresponding components
Subtraction (A - B): Subtract corresponding components
Scalar Multiplication (k × A): Multiply each component by a scalar

Vector Properties

Calculate important properties of a single vector:

Magnitude (|v|): The length of the vector: √(x² + y² + z²)
Unit Vector: A vector of length 1 in the same direction: v/|v|

Dot & Cross Products

Calculate products between two vectors:

Dot Product (A · B): A scalar equal to |A||B|cos(θ). Result is a number.
Cross Product (A × B): A vector perpendicular to both A and B (3D only)
Angle: The angle between the two vectors in degrees

Vector Projection

Project vector A onto vector B. The projection is the component of A that lies along the direction of B. Formula: proj_B(A) = (A · B / |B|²) × B

What is a Vector?

A vector is a quantity with both magnitude (size) and direction. Vectors are represented as ordered lists of numbers called components. In 2D: (x, y), in 3D: (x, y, z).

Common Applications

Physics (force, velocity, acceleration)
Computer graphics and game development
Engineering calculations
Machine learning and data science
Navigation and GPS systems

Key Formulas

Magnitude: |v| = √(x² + y² + z²)
Dot Product: A · B = a₁b₁ + a₂b₂ + a₃b₃
Cross Product: A × B = (a₂b₃ - a₃b₂, a₃b₁ - a₁b₃, a₁b₂ - a₂b₁)