Vector Addition Calculator
Add vectors in 2D or 3D using Cartesian coordinates or polar representation
Vector Addition & Subtraction
Vector A
Vector B
Result Vector
Cartesian Coordinates
Polar Coordinates
Step-by-step Calculation
Formula: A + B
Vector A: (0.00, 0.00)
Vector B: (0.00, 0.00)
Calculation: (0.00, 0.00) + (0.00, 0.00)
Result: (0.000, 0.000)
Vector Analysis
Example Calculation
2D Vector Addition Example
Vector A: (3, 4)
Vector B: (1, 2)
Operation: A + B
Calculation: (3 + 1, 4 + 2) = (4, 6)
Magnitude: √(4² + 6²) = √52 = 7.21
Angle: arctan(6/4) = 56.31°
3D Vector Subtraction Example
Vector A: (5, 3, 2)
Vector B: (2, 1, 4)
Operation: A - B
Calculation: (5 - 2, 3 - 1, 2 - 4) = (3, 2, -2)
Magnitude: √(3² + 2² + (-2)²) = √17 = 4.12
Vector Properties
Magnitude
Length of the vector
|v| = √(x² + y² + z²)
Direction
Angle from positive x-axis
θ = arctan(y/x)
Unit Vector
Vector with magnitude 1
û = v / |v|
Coordinate Conversion
Polar to Cartesian
y = r × sin(θ)
Cartesian to Polar
θ = arctan(y/x)
Vector Tips
Vector addition is commutative: A + B = B + A
Vector addition is associative: (A + B) + C = A + (B + C)
The zero vector (0, 0) is the additive identity
Vector subtraction: A - B = A + (-B)
Understanding Vector Addition
What is a Vector?
A vector is a mathematical object that has both magnitude (length) and direction. Vectors can be represented as ordered sequences of numbers (coordinates) or as arrows in space. They are fundamental in physics, engineering, and mathematics.
Vector Addition Formula
2D: (a, b) + (c, d) = (a + c, b + d)
3D: (a, b, c) + (d, e, f) = (a + d, b + e, c + f)
Vector addition is performed component-wise: simply add the corresponding coordinates of each vector to get the result.
Parallelogram Rule
Geometrically, vector addition follows the parallelogram rule. When two vectors are drawn from the same starting point, their sum is the diagonal of the parallelogram formed by the two vectors as adjacent sides.
Applications
- •Physics: Adding forces, velocities, and accelerations
- •Engineering: Structural analysis and load calculations
- •Computer Graphics: 3D transformations and animations
- •Navigation: Combining displacement vectors