Vector Addition Calculator

What is Vector Addition?

Vector addition is the process of combining two or more vectors into a single resultant vector. Unlike scalar addition, vector addition must account for both the magnitude (size) and the direction of each vector. It is a fundamental operation in physics and engineering, used to calculate net force, velocity, displacement, and more.

Vector Addition Formulas

To add vectors, we sum their corresponding components. This is mathematically straightforward and works for any number of dimensions.

2D Vector Addition

For two vectors A = (Aₓ, Aᵧ) and B = (Bₓ, Bᵧ), the resultant vector R is:

R = (Aₓ + Bₓ, Aᵧ + Bᵧ)

3D Vector Addition

For two vectors A = (Aₓ, Aᵧ, A₂) and B = (Bₓ, Bᵧ, B₂), the resultant vector R is:

R = (Aₓ + Bₓ, Aᵧ + Bᵧ, A₂ + B₂)

Magnitude and Direction

Magnitude (Length)

The magnitude of a vector v = (x, y, z) is calculated using the Pythagorean theorem:

|v| = √(x² + y² + z²)

Direction

• In 2D: The direction is often given as an angle θ from the positive x-axis, where θ = tan⁻¹(y/x).
• In 3D: The direction is typically expressed using direction cosines (the angles the vector makes with each axis).

How to Use This Calculator

1. Select Dimension: Choose between 2D (x, y) or 3D (x, y, z) mode.
2. Enter Vector Components: Input the coordinates for Vector A and Vector B.
3. Add More Vectors (Optional): Use the plus button to add additional vectors to the sum.
4. Review Results: View the resultant vector components, its magnitude, and direction indicators.

Applications of Vector Addition

• Force Vectors: Calculating the net force acting on an object.
• Velocity: Determining ground speed and heading of an aircraft in wind.
• Navigation: Calculating displacement from multiple travel legs.
• Computer Graphics: Positioning and moving objects in 3D environments.

Frequently Asked Questions

Does the order of vector addition matter?

No. Vector addition is commutative (A + B = B + A) and associative ((A + B) + C = A + (B + C)).

What is a negative vector?

A negative vector has the same magnitude but points in the opposite direction. Subtracting a vector is the same as adding its negative.

How accurate are the magnitude calculations?

The magnitude calculations use double-precision floating-point math, providing accuracy to many decimal places.

Note: Vectors can be visualized using the 'head-to-tail' method or the parallelogram rule. This calculator provides the analytical solution using component addition.