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.
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.
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.
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.
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.
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.
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.
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.
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.