Vector Addition Calculator

A vector is a mathematical object that has both magnitude (size) and direction. It is represented as an ordered set of numbers called components — two components (x, y) in 2D and three components (x, y, z) in 3D. Vectors are widely used in physics, engineering, and computer graphics to represent quantities like force, velocity, and displacement.

Vector addition is done component-by-component. For two 2D vectors A = (a, b) and B = (d, e), their sum is R = (a + d, b + e). In 3D, for A = (a, b, c) and B = (d, e, f), the result is R = (a + d, b + e, c + f). This calculator handles both cases automatically.

Simply add the corresponding components: x-component = 2 + 1 = 3, y-component = 1 + 0 = 1. The resultant vector is (3, 1). Its magnitude is √(3² + 1²) = √10 ≈ 3.1623, and its direction angle is arctan(1/3) ≈ 18.43°.

The magnitude is calculated using the Pythagorean theorem extended to your number of dimensions. In 2D: |R| = √(Rx² + Ry²). In 3D: |R| = √(Rx² + Ry² + Rz²). This calculator computes and displays the magnitude automatically after you enter the vector components.

In 2D, the direction angle from the positive x-axis is found using θ = arctan(Ry / Rx), adjusted for the correct quadrant. In 3D, two angles are typically given: the azimuthal angle in the xy-plane and the elevation angle above the xy-plane. This calculator provides both.

A scalar multiple scales all components of a vector by a constant factor before the addition is performed. For example, if you set the scalar for Vector A to 2 and enter A = (3, 1), the calculator uses (6, 2) in the addition. Setting a scalar to –1 effectively subtracts that vector, making this tool useful for vector subtraction as well.

The parallelogram rule is a geometric method for adding two vectors. You place both vectors tail-to-tail, then construct a parallelogram using them as adjacent sides. The diagonal of the parallelogram drawn from the common tail represents the resultant vector in both magnitude and direction.

Yes. To subtract Vector B from Vector A, simply set the scalar multiple for Vector B to –1. The calculator will then compute A + (–1)×B = A – B, giving you the difference vector along with its magnitude and direction.