Vector Addition Calculator
Add two vectors component by component. Enter the x, y and z values of vector a and vector b to get the resultant a + b, the magnitude of the resultant, and its direction.
Example: with a: x 3 · a: y 4 · a: z 0 · b: x 1 · b: y 2 → Resultant a + b: (4, 6, 0).
- Magnitude of resultant7.2111
- Direction56.3099° from +x axis
Computed by the calculator below using its default values. Change any input to see your own numbers.
Vectors add component by component: (ax+bx, ay+by, az+bz). Geometrically this is the tip-to-tail rule — the resultant runs from the start of the first vector to the tip of the second.
Tip-to-tail, component by component
Adding vectors is simpler than it looks: line up the components and add them. The x of the sum is the sum of the x values, and the same for y and z. Geometrically this is the tip-to-tail rule — slide the second vector so its tail meets the first vector tip, and the resultant reaches from the very start to the very end.
The magnitude of the sum is generally not the sum of the magnitudes. Two vectors of length 5 can add to anything from 0 (pointing opposite) to 10 (pointing the same way); the default here, |a+b| = 7.21, sits in between because the vectors are at an angle.
How it’s calculated
Component sum: a + b = (ax+bx, ay+by, az+bz). The resultant magnitude is √(Rx²+Ry²+Rz²). When both z components are 0 the direction is θ = atan2(Ry, Rx) from the +x axis; otherwise the direction is given as direction cosines (Rx/|R|, Ry/|R|, Rz/|R|).
Standard Cartesian components; subtraction is the same operation with the second vector negated.
Sample vector sums
| a | b | a + b | |a + b| |
|---|---|---|---|
| (3, 4) | (1, 2) | (4, 6) | 7.21 |
| (1, 0) | (0, 1) | (1, 1) | 1.41 |
| (5, 0) | (0, 5) | (5, 5) | 7.07 |
| (2, 3) | (-2, -3) | (0, 0) | 0 |
Computed by adding components, then |R| = √(Rx²+Ry²).
Common mistakes
- Adding the magnitudes instead of the components — |a+b| is not |a|+|b| unless the vectors are parallel.
- Mismatching components, for example adding a-x to b-y.
- Ignoring signs: a negative component subtracts, which can shrink the resultant.
- Reading direction in degrees when the rest of a problem uses radians.
Frequently asked questions
How do you add two vectors?
Add them component by component: (ax+bx, ay+by, az+bz). The result is the resultant vector, which points from the start of the first vector to the tip of the second.
Is the magnitude of the sum the sum of the magnitudes?
Only when the vectors point the same direction. Otherwise |a+b| is smaller, found from the resultant components as √(Rx²+Ry²+Rz²).
How do I subtract vectors instead?
Negate the second vector and add: a − b = (ax−bx, ay−by, az−bz). Everything else works the same.
What is the direction of the resultant?
For a 2D sum it is the angle from the positive x-axis, atan2(Ry, Rx). For a 3D sum this tool reports direction cosines instead.