Vector Addition Calculator
Add or subtract vectors a and b by matching components. Choose the operation, enter ax through bz, and read the result triple plus its Euclidean magnitude.
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How to add or subtract vectors
- Choose Add or Subtract in the operation control. Add forms a + b. Subtract forms a − b.
- Enter vector a as ax, ay, and az. Defaults are 1, 2, and 3.
- Enter vector b as bx, by, and bz. Defaults are 4, 5, and 6.
- Read result x, y, and z. Default add yields (5, 7, 9).
- Read the magnitude of that result vector when you need its length in the same step.
Vector addition and subtraction
Component-wise sum or difference of two Cartesian vectors
| Add rule | (ax + bx, ay + by, az + bz) |
|---|---|
| Subtract rule | (ax − bx, ay − by, az − bz) |
| Default add | (1, 2, 3) + (4, 5, 6) = (5, 7, 9) |
| Result magnitude | √(rx² + ry² + rz²) for the chosen result |
| Dimension | Up to three components; set unused z values to 0 for 2D |
| Privacy | Arithmetic runs in this tab. CZNull does not receive the numbers. |
What component-wise addition does
Vector addition lines up matching axes and sums them. Subtraction keeps a and subtracts each b component. Tip-to-tail geometry in the plane is the same algebra when you write the arrows as components. Defaults (1, 2, 3) and (4, 5, 6) add to (5, 7, 9).
How this page forms the result
For Add, each result axis is a + b on that axis. For Subtract, each result axis is a − b. All six components must be finite numbers. The page then reports √(rx² + ry² + rz²) for the result so you can see length without a second tool.
Order and commutativity
Addition is commutative: a + b equals b + a. Subtraction is not: a − b differs from b − a unless the vectors match. Keep a as the left operand when you choose Subtract.
Use cases
Combining force components, stacking displacements in physics labs, and checking graphics offsets. Polar form with magnitude and angle needs conversion to components before you use this form.
Limits
Component-wise only. No matrix transforms. No more than three dimensions. IEEE floating point applies. See the disclaimer.