What is Vector Addition?

Vector addition is the process of finding a single resultant vector that represents the combined effect of two or more individual vectors. Unlike simple scalar addition, vector addition must account for both magnitude (size) and direction. In a Cartesian coordinate system, this is most easily achieved by adding the corresponding components (x, y, and z) of each vector.

How to Use This Vector Calculator

To use our vector addition calculator with steps, simply enter the coordinates for Vector A and Vector B in the provided fields. You can enter values for 2D vectors (leaving Z as zero) or 3D vectors. Once you click "Calculate Resultant," the tool will provide the sum of the vectors and show you exactly how the math was performed, including the calculation of the magnitude (length) of the resulting vector.

The Formula for Adding Vectors

If you have two vectors A = (x₁, y₁, z₁) and B = (x₂, y₂, z₂), their sum R is calculated as:

R = (x₁ + x₂, y₁ + y₂, z₁ + z₂)

The magnitude of the resultant vector is found using the Pythagorean theorem extended to three dimensions:

|R| = √(Rx² + Ry² + Rz²)

Frequently Asked Questions

Can I add vectors with different dimensions?
No, vectors must have the same number of dimensions to be added together (e.g., you cannot add a 2D vector to a 3D vector directly).

Is vector addition commutative?
Yes, vector addition follows the commutative law, meaning A + B = B + A. The order in which you add the vectors does not change the final resultant vector.

What is a resultant vector?
The resultant vector is the algebraic sum of two or more vectors. It represents the net displacement or force when multiple vectors act simultaneously.