Vector Addition Calculator
Reviewed by Hamza Gujjar, Algorithm Architect & Software Developer
Last updated: August 19, 2026Combining Two Directions Into One
Adding vectors combines two separate magnitudes-and-directions — two velocities, two forces, two displacements — into the single net effect they produce together. Unlike ordinary number addition, the result depends on direction as much as size, but the arithmetic itself stays simple: add each matching component separately.
The Formula
3D: A + B = (Ax+Bx, Ay+By, Az+Bz)
Worked Example
Adding vector A = (3, 4) and vector B = (1, 2):
| Component | Calculation | Result |
|---|---|---|
| x | 3 + 1 | 4 |
| y | 4 + 2 | 6 |
Resulting vector: (4, 6).
Where This Matters
- Force analysis — combining multiple forces acting on an object to find the net force, a core step in statics and structural engineering.
- Navigation — adding a vehicle's velocity vector to a wind or current vector to find actual ground-track speed and direction.
- Game physics — summing movement, gravity, and impulse vectors each frame to update an object's position.
How to Use This Calculator
- Choose the dimension: 2D Vectors or 3D Vectors.
- Enter Vector A's x, y (and z for 3D) components.
- Enter Vector B's x, y (and z for 3D) components.
- Select Calculate to get the resultant vector.
Related Calculations
Need the length of the result? Use the Vector Magnitude Calculator, or find the angle between two vectors with the Angle Between Vectors Calculator.
Principles of Vector Addition and Geometric Superposition
In linear algebra, classical mechanics, and physics, a vector is a geometric entity possessing both magnitude (length) and spatial direction. Vector addition is the algebraic and geometric operation of combining two or more individual vectors into a single Resultant Vector (R). Unlike scalar addition where magnitudes simply sum arithmetically, vector addition accounts for the relative spatial angles between vectors.
Geometric Laws: The Parallelogram and Triangle Methods
- The Triangle Law (Head-to-Tail Method): To add vector v to vector u, place the tail (origin) of vector v at the head (arrowhead) of vector u. The resultant vector R = u + v is the directed line segment drawn from the tail of u directly to the head of v.
- The Parallelogram Law: When two concurrent vectors originate from a common point, their sum forms the diagonal of the parallelogram constructed with the two vectors as adjacent sides:
||R|| = √[ ||u||² + ||v||² + 2 × ||u|| × ||v|| × cos(θ) ]
Algebraic Component-Wise Vector Addition
In Cartesian coordinate systems, vector addition is performed by adding corresponding orthogonal components independently across Euclidean space:
3D Addition: 〈ux, uy, uz〉 + 〈vx, vy, vz〉 = 〈ux + vx, uy + vy, uz + vz〉
Step-by-Step Worked Calculation Example
Example: Aviation Wind Triangle Navigation Vector Addition
Problem: An aircraft flies with a True Airspeed vector u pointing directly North (Heading = 000°) at 200.0 knots: u = 〈0.0, 200.0〉 knots. A steady crosswind vector w blows from the West toward the East (Bearing = 090°) at 45.0 knots: w = 〈45.0, 0.0〉 knots. Calculate: (1) The resultant Groundspeed vector R; (2) The actual Groundspeed magnitude; and (3) The aircraft's true Ground Track drift angle (θ).
Step 1: Perform component-wise addition:
Rx = ux + wx = 0.0 + 45.0 = 45.0 knots (Eastward drift)
Ry = uy + wy = 200.0 + 0.0 = 200.0 knots (Northward progress)
Resultant Vector R = 〈45.0, 200.0〉 knots
Step 2: Calculate resultant Groundspeed magnitude:
||R|| = √[ Rx² + Ry² ] = √[ 45.0² + 200.0² ] = √[ 2,025 + 40,000 ] = √42,025 = 205.00 knots
Step 3: Calculate the wind drift angle (θ):
θ = arctan(Rx / Ry) = arctan(45.0 / 200.0) = arctan(0.225) = 12.68° East of North (Track = 012.7°)
Conclusion: The aircraft travels over the ground at 205.0 knots along an actual ground track of 012.7°.
Engineering Applications: Static Equilibrium
In civil structural engineering, a truss joint or bridge cable anchor is in static equilibrium if and only if the vector sum of all concurrent applied forces equals zero:
Linear Combinations and Vector Space Axioms
In abstract linear algebra, vector addition is one of the two foundational operations defining a formal Vector Space (V, +, ·) over a field 𝔽 (such as the real numbers ℝ). For any vectors u, v, w ∈ V and scalars c, d ∈ 𝔽, vector addition satisfies eight rigorous algebraic axioms:
- Closure under Addition: u + v ∈ V.
- Commutativity: u + v = v + u.
- Associativity: (u + v) + w = u + (v + w).
- Additive Identity: Existence of a unique zero vector 0 such that v + 0 = v.
- Additive Inverse: Existence of unique inverse -v such that v + (-v) = 0.
Vector Spanning Sets and Linear Independence
A Linear Combination of a set of vectors {v1, v2, …, vk} is any vector formed by multiplying each vector by a scalar weight and adding the results:
The set of all possible linear combinations forms the Span of the vectors: Span{v1, …, vk}. If the only solution to c1v1 + … + ckvk = 0 is c1 = c2 = … = 0, the vectors are Linearly Independent, forming a coordinate basis for the vector space.
Relativistic Velocity Addition in Special Relativity
In Albert Einstein's Special Theory of Relativity, when velocities approach the speed of light c (approx. 300,000 km/s), classical Newtonian Galilean vector addition breaks down. Relativistic collinear velocity addition incorporates the Lorentz velocity transformation:
This ensures that no compounded velocity can ever exceed the universal cosmic speed of light c.
Polygon Law of Vector Addition for Multi-Vector Systems
When adding three or more vectors concurrently (R = v1 + v2 + v3 + … + vn), the geometric head-to-tail triangle law generalizes into the Polygon Law of Vector Addition. Placing each subsequent vector's tail at the preceding vector's head forms an open polygon chain; the resultant vector R is the closing directed segment drawn from the initial origin to the final terminal tip. If the polygon closes onto its own origin, the net resultant is the zero vector (R = 0), confirming that the multi-body physical system is in complete static translational equilibrium.
Vector Subtraction as Inverse Addition
Vector subtraction is algebraically defined as the addition of the negative vector: u - v = u + (-v). Geometrically, if vector u and vector v share a common origin, the difference vector d = u - v is the directed line segment drawn from the head of v to the head of u, representing relative displacement in kinematics.
Affine Spaces and Point-Vector Addition
In computational geometry and computer-aided design (CAD), geometric points P and directional vectors v inhabit an Affine Space. Adding a vector to a point produces a new translated point (P' = P + v), while subtracting two points yields the directed displacement vector between them (v = Q - P), forming the foundation of parametric Bezier curves and 3D spline modeling.