Vector Addition

Calculator and formulas for adding 2D, 3D, and 4D vectors

Vector Calculator

Component-wise Vector Addition

This calculator can add vectors of different dimensions by performing component-wise addition calculations.

Select Vector Dimension
First Vector (v₁)
Second Vector (v₂)
Addition Result
X:
Y:
Z:
W:
Components are added individually: v₁ + v₂ = [x₁+x₂, y₁+y₂, ...]

Vector Addition Info

Vector Properties

Component-wise: Corresponding components are added

Commutative Associative Zero Vector

Prerequisite: Same number of components
Result: New vector of same dimension

Example
[2, 4, 1] + [3, 5, 2] = [5, 9, 3]
[1, -2] + [3, 4] = [4, 2]
[0, 0, 0] + [x, y, z] = [x, y, z]

Formulas for Vector Addition

2D Vector Addition
\[\left[\matrix{a\\b}\right] + \left[\matrix{c\\d}\right] = \left[\matrix{a+c\\b+d}\right]\]

Addition in the plane

3D Vector Addition
\[\left[\matrix{a\\b\\c}\right] + \left[\matrix{x\\y\\z}\right] = \left[\matrix{a+x\\b+y\\c+z}\right]\]

Addition in space

4D Vector Addition
\[\left[\matrix{a\\b\\c\\d}\right] + \left[\matrix{w\\x\\y\\z}\right] = \left[\matrix{a+w\\b+x\\c+y\\d+z}\right]\]

Higher-dimensional addition

General Rule
\[\vec{v_1} + \vec{v_2} = \sum_{i} (v_{1i} + v_{2i})\vec{e_i}\]

Component-wise addition

Calculation Examples for Vector Addition

Example 1: 3D Addition
[2, 4, 1] + [3, 5, 2]
\[\vec{v_1} + \vec{v_2} = \left[\matrix{2+3\\4+5\\1+2}\right] = \left[\matrix{5\\9\\3}\right]\]

Result: [5, 9, 3]

Example 2: 2D Addition
[1, -3] + [-2, 5]
\[\vec{u} + \vec{w} = \left[\matrix{1+(-2)\\-3+5}\right] = \left[\matrix{-1\\2}\right]\]

Result: [-1, 2]

Step-by-Step Calculation
X: 2 + 3 = 5
Y: 4 + 5 = 9
Z: 1 + 2 = 3

Each component is added separately

Applications of Vector Addition

Vector addition is fundamental in many scientific and technical fields:

Physics & Mechanics
  • Force vectors and decomposition
  • Velocity and acceleration vectors
  • Electric and magnetic fields
  • Wave vectors and momentum
Computer Graphics
  • 3D object transformation
  • Animation and motion calculation
  • Lighting and shading
  • Collision detection
Navigation
  • GPS position calculation
  • Flight path and route planning
  • Wind and flow correction
  • Coordinate transformation
Robotics
  • Robot arm control
  • Path planning and motion
  • Sensor data processing
  • Inverse kinematics

Vector Addition: Foundation of Linear Algebra

Vector addition is a fundamental operation of linear algebra that adds vectors component-wise. Together with scalar multiplication, this operation forms the basis for vector spaces and enables the mathematical description of directions, forces, and movements in multidimensional spaces.

Summary

Vector addition combines geometric intuition with algebraic precision. The simple rule - add corresponding components - enables complex calculations in physics, computer graphics, and engineering. From 2D graphics through 3D modeling to high-dimensional data analysis, vector addition remains an indispensable mathematical tool. It shows how elementary operations form the foundation for advanced scientific and technical applications.

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Vector Functions

Addition • Subtraction • Multiplication • Scalar Multiplication • Division • Scalar Division • Dot Product • Cross Product • Interpolation • Distance • Distance Squaret • Normalization • Reflection • Magnitude • Squared-Magnitude • Triple-Product