Vector addition: to add two vectors, add their components and recombine — Rx = Ax + Bx, Ry = Ay + By, |R| = sqrt(Rx² + Ry²). This free calculator takes magnitudes and angles or x/y components and returns the resultant’s magnitude, direction and components, with every step shown.
What Is the Vector Addition Calculator?
The vector addition calculator is a free online tool built on the formula R = sqrt(Rx² + Ry²). Enter the values you already know, in whichever units suit you, and it solves for resultant magnitude, direction and components and shows every step of the substitution. Vector Addition is add two vectors from magnitudes & angles or x/y components to get the resultant.
| Symbol | Quantity | Default unit | Also accepts | Example value |
|---|---|---|---|---|
| |A| | Vector A magnitude | units | — | 3 |
| θA | Vector A angle | ° | — | 0 |
| |B| | Vector B magnitude | units | — | 4 |
| θB | Vector B angle | ° | — | 90 |
| Ax | A x-component | units | — | — |
| Ay | A y-component | units | — | — |
| Bx | B x-component | units | — | — |
| By | B y-component | units | — | — |
How to calculate the resultant of two vectors
Two vectors do not add like ordinary numbers — a 3-unit push east and a 4-unit push north combine to a 5-unit resultant, not 7. The reliable method is to work in components. Resolve each vector onto the x and y axes using Ax = |A|·cos θ_A and Ay = |A|·sin θ_A, where the angle θ is measured anticlockwise from the positive x-axis. Do the same for B.
Now add the like components: Rx = Ax + Bx and Ry = Ay + By. These two numbers fully describe the resultant. To express it as a magnitude and direction, use Pythagoras and the arctangent: |R| = sqrt(Rx² + Ry²) and φ = atan2(Ry, Rx). The atan2 function is used instead of a plain tan-¹ because it returns the correct quadrant and never divides by zero — even when Rx = 0.
If you already know the components, switch the calculator to Components mode and type Ax, Ay, Bx and By directly. Either way the engine adds the components and reports |R|, the direction (both as a ±180° value and as a 0–360° value), and Rx and Ry. For the underlying ideas, see the physics glossary.
Worked example
Add A = 3 units at 0° (due east) and B = 4 units at 90° (due north). The components are Ax = 3, Ay = 0, Bx = 0, By = 4, so Rx = 3 + 0 = 3 and Ry = 0 + 4 = 4. The magnitude is |R| = sqrt(3² + 4²) = sqrt(25) = 5.00 units and the direction is φ = atan2(4, 3) = 53.13° from the +x axis. (At three significant figures the tool shows 5 and 53.1°.) This is the classic 3-4-5 right triangle: the resultant is shorter than 3 + 4 = 7 because the two vectors are perpendicular, not aligned.
Why it matters
Adding vectors is the first real skill of mechanics. Forces on a bridge joint, the velocity of a plane in a crosswind, the displacement of a hiker after several legs, the net field from two charges — all are vector sums. Get the component method right here and the same recipe carries through statics, kinematics, electromagnetism and beyond.
Frequently asked questions
How do you add two vectors?
Resolve each vector into x and y components, add the components separately, then recombine. With Ax = |A|cos θ_A and Ay = |A|sin θ_A (and likewise for B), the resultant is Rx = Ax + Bx, Ry = Ay + By, with magnitude |R| = sqrt(Rx² + Ry²) and direction φ = atan2(Ry, Rx). You cannot simply add the magnitudes unless the vectors point the same way.
What is the difference between magnitude-angle and component input?
They are two descriptions of the same vector. Magnitude-and-angle gives a length and a direction (for example 5 units at 53°); components give the x and y parts (for example 3 and 4). This calculator accepts either — Ax = |A|cos θ and Ay = |A|sin θ convert one to the other — and you can even describe A one way and B the other.
How is the direction angle measured?
The direction φ is measured anticlockwise from the positive x-axis. The result is reported two ways: as a signed angle in the range −180° to +180° (the natural output of atan2), and as a positive angle from 0° to 360°. Both describe the same direction.
What happens when two equal and opposite vectors are added?
The components cancel, so the resultant is the zero vector: Rx = 0, Ry = 0 and |R| = 0. A zero vector has no direction, so the calculator reports the magnitude as 0 and the direction as undefined rather than printing a meaningless angle.
Why must the magnitudes be non-negative?
A magnitude is a length, and lengths cannot be negative. A “negative magnitude” really means the same length pointing in the opposite direction — a 180° change of angle. So enter a non-negative magnitude and set the angle to control the direction; the calculator rejects negative magnitudes to avoid ambiguity.
References & formula source
- Halliday, Resnick & Walker — Fundamentals of Physics, Chapter 3 (Vectors).
- Young & Freedman — University Physics with Modern Physics, Chapter 1 (Units, Physical Quantities and Vectors).
- NASA Glenn Research Center — Beginner’s Guide to Aeronautics: Scalars and Vectors / Vector Addition.
- Further reading: Euclidean vector — Wikipedia