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

(0.000, 0.000)
x: 0.000, y: 0.000

Polar Coordinates

Magnitude: 0.000
Angle: 0.0°
r = 0.000, θ = 0.0°

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)

θ

Direction

Angle from positive x-axis

θ = arctan(y/x)

û

Unit Vector

Vector with magnitude 1

û = v / |v|

Coordinate Conversion

Polar to Cartesian

x = r × cos(θ)
y = r × sin(θ)

Cartesian to Polar

r = √(x² + y²)
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