Add two 2D vectors by combining their x and y components.
How it works
Vector addition is component-wise: (ax, ay) + (bx, by) = (ax + bx, ay + by). The vectors (2, 3) and (4, -1) add to (6, 2).
What this does not include
This is distinct from this site’s angle-between-two-vectors calculator (which finds an angle, not a sum) and vector magnitude calculator (a single vector’s length). This handles 2D vectors only.
How to use this calculator
- Enter the x and y components of vector a.
- Enter the x and y components of vector b.
A worked example
Vector A (2, 3) plus vector B (4, −1): sum = (6, 2) — add each component separately.
Vector A (−1, 5) plus vector B (3, 2): sum = (2, 7).
What the variables mean
| Variable | Meaning |
|---|---|
| Vector A (ax, ay) | First vector’s components |
| Vector B (bx, by) | Second vector’s components |
Edge cases worth knowing
Vector addition is just component-by-component addition — x-components add to x-components, y-components add to y-components, entirely independent of each other.
Order doesn’t matter — A + B always equals B + A for vector addition, unlike some other vector operations like the cross product.
Why does vector addition work component-wise?
Each component represents movement along one independent direction (x or y), so combining two vectors just means combining how far each moves along each of those directions separately.
Does the order of addition matter?
No — vector addition is commutative, so a + b gives the same result as b + a, just like ordinary number addition.
What does the resulting vector represent geometrically?
It’s the vector you’d get by placing the tail of vector b at the tip of vector a (or vice versa) — the “tip-to-tail” method taught in physics and geometry classes.
