What this calculator does
Vector addition combines two vectors by adding their matching components: the x-components add together, the y-components add together, and the z-components add together if working in three dimensions. The result is a single new vector, called the resultant, that represents the combined effect of the two originals.
This is a genuinely different operation from finding a vector's length or the angle between two vectors: addition produces a new vector, not a single number. It comes up whenever two quantities that have both size and direction need to be combined, such as two forces acting on an object, or two displacements applied one after another.
The formula
Add each pair of matching components separately: the resultant's x-component is the sum of the two x-components, and likewise for y and z. The magnitude of that resultant vector is then found the usual way, by taking the square root of the sum of the squared components. Leave the z fields at zero to work entirely in two dimensions.
| Term | Meaning |
|---|---|
| Component | The part of a vector along one axis, such as its x-value or y-value. |
| Resultant vector | The single vector produced by adding two (or more) vectors together. |
| Magnitude | The length of a vector, found from its components as √(x² + y² + z²), regardless of direction. |
The inputs explained
| Field | What to enter |
|---|---|
| Vector A, x | The x-component of the first vector. |
| Vector A, y | The y-component of the first vector. |
| Vector A, z (0 for 2D) | The z-component of the first vector. Leave at 0 for a 2D vector. |
| Vector B, x | The x-component of the second vector. |
| Vector B, y | The y-component of the second vector. |
| Vector B, z (0 for 2D) | The z-component of the second vector. Leave at 0 for a 2D vector. |
When to use it
Combining two forces
Two forces acting on the same object at different angles do not simply add as plain numbers; resolving each into x and y components first, then adding those components, gives the true combined force as a resultant vector.
Adding successive displacements
Walking or travelling along one path and then another is the sum of two displacement vectors; the resultant vector gives the straight-line net displacement, not the total distance actually travelled.
Combining velocities
A boat's velocity through water added to the water current's velocity gives the boat's actual velocity over the ground, found the same way by summing the two vectors' components.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
Adding a fixed vector A = (3, 4) to vector B as its x-component grows
Vector A held fixed at (3, 4) and vector B's y-component fixed at 2, with B's x-component stepped up to show how the resultant and its magnitude change.
| Vector B x-component | Resultant vector | Magnitude of resultant |
|---|---|---|
| 1 | (4.000, 6.000) | 7.211 |
| 2 | (5.000, 6.000) | 7.810 |
| 3 | (6.000, 6.000) | 8.485 |
| 4 | (7.000, 6.000) | 9.220 |
| 5 | (8.000, 6.000) | 10.000 |
| 6 | (9.000, 6.000) | 10.817 |
Adding two 3D vectors
A fixed unit vector along x, added to a vector B whose z-component increases while its x and y stay at zero, to show a genuinely three-dimensional sum.
| Vector B z-component | Resultant vector | Magnitude of resultant |
|---|---|---|
| 0 | (1.000, 0.000) | 1.000 |
| 1 | (1.000, 0.000, 1.000) | 1.414 |
| 2 | (1.000, 0.000, 2.000) | 2.236 |
| 3 | (1.000, 0.000, 3.000) | 3.162 |
| 4 | (1.000, 0.000, 4.000) | 4.123 |
| 5 | (1.000, 0.000, 5.000) | 5.099 |
Questions
Does the order the vectors are added in matter?
No. Vector addition is commutative, so A + B gives exactly the same resultant as B + A, the same way ordinary addition of numbers does.
How do I add more than two vectors?
Add the first two to get a resultant, then add that resultant to the next vector, and repeat. Each step uses the same component-by-component addition this calculator performs, so a chain of vectors reduces to a series of two-vector additions.
Is this the same as vector subtraction?
Subtraction is addition of the negative: to subtract vector B from vector A, add A to a version of B with every component flipped in sign. The mechanics are otherwise identical.
What is the difference between this and finding the angle between two vectors?
Vector addition produces a new vector (the resultant), found by adding components. The angle between two vectors is a single number, found from the dot product, and does not involve creating any new vector at all.
To find the length of a single vector rather than a sum of two, use the vector magnitude calculator. For the angle between two vectors instead of their sum, see the angle between two vectors calculator.