Vector Addition Calculator — 2D Resultant, Magnitude & Direction
Enter the x and y components of two (or three) 2D vectors to instantly compute the resultant vector: its x/y components, magnitude, and direction angle — with an animated arrow diagram.
Magnitude of the vector sum A + B (+ C if given)
- 1
Sum of x components (Rx)
3 + 1 = 4 - 2
Sum of y components (Ry)
4 + -2 = 2 - 3
Resultant magnitude |R|
√(4² + 2²) = 4.4721The Euclidean length of the resultant vector.
How does this calculator work?
Add 2D vectors component-by-component: Rx = Ax + Bx, Ry = Ay + By. Resultant magnitude |R| = √(Rx² + Ry²); direction θ = atan2(Ry, Rx). Supports up to 3 vectors. Order does not matter — vector addition is commutative.
Formula
How this is calculated
Vector addition in two dimensions is performed component-by-component: add all x-components to get Rx and all y-components to get Ry. The resultant vector R = (Rx, Ry) represents the single vector with the same combined effect as applying A, B (and optionally C) in sequence — this is the triangle or parallelogram rule made arithmetic.
The magnitude of the resultant is the Euclidean length |R| = √(Rx² + Ry²), which equals the hypotenuse of the right triangle formed by the two components. The direction angle θ = atan2(Ry, Rx) gives the standard position angle measured counter-clockwise from the positive x-axis, in degrees. This is the standard mathematical convention; bearing angles in navigation add 90° and measure clockwise from north.
The diagram draws each component vector as an arrow from the origin and the resultant in green. This visualisation corresponds to the tip-to-tail (polygon) method of vector addition: if you placed the tail of B at the tip of A, the line from the original origin to the final tip of B would be the resultant.
Frequently asked questions
θ is the angle of the resultant vector measured counter-clockwise from the positive x-axis (standard mathematical convention). A vector pointing due right has θ = 0°, due up has θ = 90°, due left has θ = 180°, and due down has θ = −90° (or equivalently 270°). In physics and engineering problems stated as "N at 30° above the x-axis", the x-component is |N|cos(30°) and the y-component is |N|sin(30°).
No — vector addition is commutative (A + B = B + A) and associative ((A + B) + C = A + (B + C)). The resultant is the same regardless of the order in which the vectors are added. Geometrically, the tip-to-tail drawing changes shape depending on order, but the resultant from origin to final tip is always the same vector.
If you know a vector's magnitude |v| and direction angle θ (from the positive x-axis), the components are: x = |v| × cos(θ) and y = |v| × sin(θ). For example, a force of 10 N at 30° has x = 10 cos(30°) ≈ 8.66 N and y = 10 sin(30°) = 5 N. Enter these resolved components into the calculator to add multiple vectors given as magnitude + angle.
Also known as
TG we-Calculate Editorial Team. (2026). Vector Addition Calculator — 2D Resultant, Magnitude & Direction [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/vector-addition-calculator
TG we-Calculate Editorial Team. "Vector Addition Calculator — 2D Resultant, Magnitude & Direction." TG we-Calculate. 2026. https://we-calculate.com/calculator/vector-addition-calculator.
TG we-Calculate Editorial Team, "Vector Addition Calculator — 2D Resultant, Magnitude & Direction," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/vector-addition-calculator
@misc{wecalculate_vector_addition_calculator, title = {Vector Addition Calculator — 2D Resultant, Magnitude & Direction}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/vector-addition-calculator}}, year = {2026}, note = {TG we-Calculate} }
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