Adding vectors combines displacements, forces and velocities: a plane’s airspeed plus the wind gives its ground speed. The parallelogram (or tip-to-tail) rule is the geometric picture of component-wise addition.
How to use the Vector Addition Calculator
- Vectors in
- Operation
- Vector representation (2D)
- Multiple α (for a)
- Multiple β (for b)
- a – x (or magnitude)
- a – y (or angle °)
- a – z
- b – x (or magnitude)
- b – y (or angle °)
- b – z
Results update as you type, and everything is calculated in your browser.
How it works
Vectors given as magnitude and angle are converted to components first. The result is shown as components, magnitude and direction.
Tips
- Sketch the points first – a quick drawing shows whether a slope should be positive or negative.
- Keep the order of the points consistent in every formula.
- Vertical lines have an undefined slope; horizontal lines have slope zero.
- Use the same units on both axes when you measure distances or angles.
Common mistakes to avoid
- Swapping x and y coordinates.
- Subtracting the coordinates in a different order in the numerator and denominator of the slope.
- Forgetting that the distance is always positive.
Inputs explained
| Field | What to enter |
|---|---|
| Vectors in | Choose one: 2D, 3D |
| Operation | Choose one: Addition, Subtraction |
| Vector representation (2D) | Choose one: Cartesian coordinates, Magnitude and angle |
| Multiple α (for a) | A number |
| Multiple β (for b) | A number |
| a – x (or magnitude) | A number |
| a – y (or angle °) | A number |
| a – z | A number |
| b – x (or magnitude) | A number |
| b – y (or angle °) | A number |
| b – z | A number |
Example values
How the main result changes when Multiple α changes and everything else stays at the example values above:
| Multiple α (for a) | Result – x |
|---|---|
| 0.5 | 2.5 |
| 0.75 | 3.25 |
| 1 | 4 |
| 1.25 | 4.75 |
| 1.5 | 5.5 |
| 2 | 7 |
Formula
α·a ± β·b = (αaₓ ± βbₓ, αa_y ± βb_y, αa_z ± βb_z)Polar input: x = m cos θ, y = m sin θWorked example
Inputs — Vectors in: 2D; Operation: Addition; Vector representation (2D): Cartesian coordinates; Multiple α (for a): 1; Multiple β (for b): 1; a – x (or magnitude): 3; a – y (or angle °): 4; b – x (or magnitude): 1; b – y (or angle °): -2.
Results — Result – x = 4; Result – y = 2; Magnitude of the result = 4.47213595; Direction of the result (2D) = 26.5651 °.
Key terms
- Resultant
- The vector sum of several vectors.
Frequently asked questions
How do I add vectors graphically?
Place them tip to tail; the sum goes from the first tail to the last tip.
Is vector addition commutative?
Yes, a + b = b + a.
How do I subtract vectors?
Add the opposite: a − b = a + (−b).
What do α and β do?
They scale each vector before adding – a linear combination.
What formula does the Vector Addition Calculator use?
α·a ± β·b = (αaₓ ± βbₓ, αa_y ± βb_y, αa_z ± βb_z).