Vector Addition Calculator

The Vector Addition Calculator lets you easily add, subtract, and scale vectors in both 2D and 3D space. Here is how to use it:

Step 1: Choose whether your vectors are in 2D or 3D.

Step 2: Select the operation you want to perform (Addition a + b, Subtraction a - b, or With multiples).

Step 3: Choose your preferred vector representation (Cartesian Coordinates or Magnitude and Direction).

Step 4: Enter the coordinates or magnitudes and angles for Vector a and Vector b.

Step 5: Click Calculate to find the resultant vector. The tool displays the result instantly in both Cartesian and Magnitude/Direction forms.

For vector addition in Cartesian coordinates:

If a = (a₁, a₂, a₃) and b = (b₁, b₂, b₃) are vectors in 3D, their sum is:

a + b = (a₁ + b₁, a₂ + b₂, a₃ + b₃)

For subtraction:

a − b = (a₁ − b₁, a₂ − b₂, a₃ − b₃)

When calculating with multiples (scaling vectors), the formula is:

c(a) + d(b) = (c·a₁ + d·b₁, c·a₂ + d·b₂, c·a₃ + d·b₃)

If the vectors are given as magnitude (|v|) and direction (θ), the magnitudes are first converted to Cartesian components using: x = |v| cos(θ) and y = |v| sin(θ). After the addition, the resulting vector is converted back to polar form using |R| = √(x² + y²) and θ = atan2(y, x).

The Vector Addition Calculator is a vital tool for physics students, engineers, and mathematicians dealing with geometric quantities that have both magnitude and direction. Unlike simple scalar numbers where 5 + 5 always equals 10, adding two forces of 5 Newtons can result in any value between 0 and 10 depending on the angle between them. Vector addition solves exactly this problem.

Understanding vector addition is crucial for analyzing forces in mechanics, determining net velocity in fluid dynamics, or computing electromagnetic fields. The calculation becomes increasingly tedious when moving from simple 1D forces to 2D and 3D space, which is where this tool helps by instantly resolving components and combining them accurately.

Our calculator supports multiple input formats (Cartesian (x,y) coordinates and polar (magnitude, direction) coordinates) to match the diverse ways vectors are presented in textbooks and real-world problems. It also includes functions for vector subtraction and scalar multiplication (scaling vectors by a constant factor), providing a comprehensive suite for basic vector arithmetic.