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Vector Addition Calculator

Geometry

Sample Vector Presets

Vector A

X

Y

Vector B

X

Y

Decimal Precision

Resultant Vector

(4, 6)

ABR

Magnitude

7.2111

Direction

56.3099°

0.9828 rad

Polar Form

7.2111 ∠ 56.3099°

About This Tool

Vector Addition Calculator – Resultant, Magnitude, and Direction

Vector addition is one of the most fundamental operations in physics, engineering, navigation, and computer graphics — it combines two or more vector quantities, such as forces, velocities, or displacements, into a single resultant vector. This calculator adds (or subtracts) vectors in 2D or 3D, supports combining up to six vectors at once, and reports the resultant's components, magnitude, and direction angle alongside a step-by-step breakdown.

The Component Method of Vector Addition

Vectors are added component-wise: corresponding components are simply summed together.

  • 2D: A + B = (Ax + Bx, Ay + By)
  • 3D: A + B = (Ax + Bx, Ay + By, Az + Bz)

For example, if A = (3, 4) and B = (1, 2), then A + B = (4, 6). This generalizes cleanly to more than two vectors: for any set of vectors V1, V2, …, Vn, the resultant is R = ΣVᵢ, the component-wise sum of every vector in the set — useful for superposition problems like summing several forces acting on the same object.

The Triangle and Parallelogram Rules

Geometrically, component addition is equivalent to the triangle rule: draw vector A, then draw vector B starting from the tip of A, and the resultant is the arrow from the origin to the final tip. The parallelogram ruleis an equivalent construction for two vectors — draw A and B from a shared origin, complete the parallelogram they form, and the resultant is the diagonal from that origin. This calculator's diagram illustrates the triangle rule directly, chaining each vector tip-to-tail before overlaying the resultant.

Vector Subtraction

Subtracting one vector from another, A − B, is mathematically the same as adding the negation of B: A − B = A + (−B), where every component of B is flipped in sign. For example, with A = (5, 3) and B = (2, 7), A − B = (3, −4). Vector subtraction is especially useful for finding the vector that points from one location to another, or the relative velocity between two moving objects.

Magnitude and Direction of the Resultant

Once the resultant's components are known, its length (magnitude) follows from the Pythagorean theorem: |R| = √(Rx² + Ry² [+ Rz²]). In 2D, the direction angle relative to the positive x-axis is θ = atan2(Ry, Rx), which the calculator reports in both degrees and radians, along with the equivalent polar form |R| ∠ θ.

Polar Form Input

In 2D mode, vectors can also be entered as a magnitude and angle instead of x/y components. The calculator converts each polar vector to Cartesian form using x = m·cos θ and y = m·sin θ before adding, then reports the resultant in both component and polar form — handy when your source data (like a compass bearing or a velocity reading) is naturally given as a magnitude and angle rather than x/y coordinates.

Equilibrium and the Zero Vector

Undefined direction
When a set of vectors sums to exactly zero, they are said to be in equilibrium — common in statics problems where several forces balance an object perfectly. Because a zero-length vector has no meaningful direction, this calculator reports the direction as undefined rather than an arbitrary angle whenever the resultant magnitude is zero.

Real-World Applications

  • Physics — net force: Combine multiple force vectors acting on an object to find the single net force that determines its acceleration.
  • Navigation:Add a vessel's velocity vector to a current or wind vector to find the resulting ground velocity and heading.
  • Computer graphics: Sum displacement or motion vectors to update object positions frame by frame.
  • Engineering statics: Check whether a structure is in equilibrium by verifying that all applied force vectors sum to zero.

Using This Calculator

Choose 2D or 3Dmode, enter each vector's components (or switch to polar input in 2D), and toggle between Add and Subtract for two-vector calculations. Use Add Another Vector to combine up to six vectors at once, expand the Step-by-Step Calculation panel to see the substituted component sums, and use the copy or share buttons to save your results or send a pre-filled link to a colleague or classmate.

Frequently Asked Questions

Is the Vector Addition Calculator free?

Yes, Vector Addition Calculator is totally free :)

Can I use the Vector Addition Calculator offline?

Yes, you can install the webapp as PWA.

Is it safe to use Vector Addition Calculator?

Yes, any data related to Vector Addition Calculator only stored in your browser (if storage required). You can simply clear browser cache to clear all the stored data. We do not store any data on server.

How does the Vector Addition Calculator work?

Enter the components of two or more vectors in 2D or 3D, choose Add or Subtract, and the calculator sums (or subtracts) them component-wise to produce the resultant vector, along with its magnitude and direction angle. A step-by-step breakdown shows exactly how each component was combined.

How do you add two vectors together?

Vectors are added component-wise: A + B = (Ax + Bx, Ay + By) in 2D, or (Ax + Bx, Ay + By, Az + Bz) in 3D. This is equivalent to the geometric triangle rule, where vector B is drawn starting at the tip of vector A and the resultant runs from the origin to the final tip.

How is vector subtraction different from addition?

Subtracting vector B from vector A is the same as adding the negation of B: A − B = A + (−B), where −B has every component flipped in sign. This calculator lets you toggle between Add and Subtract for two-vector calculations without re-entering values.

Can I add more than two vectors at once?

Yes, you can add up to six vectors (A through F) in a single calculation using the "Add Another Vector" control. This is useful for superposition problems, such as finding the net force from several individual forces or the total displacement from a series of movements.

What does it mean when the resultant vector is zero?

A zero resultant means the vectors cancel out exactly and are in equilibrium — for example, three forces of equal and opposite effect acting on an object. When this happens, the direction angle is mathematically undefined, and the calculator reports "undefined" instead of an arbitrary angle.

Can I enter vectors in polar form (magnitude and angle)?

Yes, in 2D mode you can switch the input format to polar and enter each vector as a magnitude and angle instead of X/Y components. The calculator automatically converts polar inputs to Cartesian components before adding them, and reports the resultant in both forms.