Free Vector Addition Calculator

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Key Features

Instant Calculation

Get accurate results in real time with our optimized algorithm.

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Vector addition combines two or more vectors into a single resultant vector by adding their corresponding components — geometrically, it's the same as placing the vectors head-to-tail and drawing an arrow from the very first tail to the very last head (the "tip-to-tail" method). This is the mathematical foundation behind combining forces in physics (finding the net force acting on an object from several individual forces), combining displacements in navigation (total displacement after several legs of a journey), and combining velocities (like a plane's airspeed vector plus a wind vector to get its ground velocity).

How it works

Choose the dimension (2D or 3D) and number of vectors (2 or 3), then enter each vector's components. The calculator adds up all the x-components to get the resultant's x-component, all the y-components for the resultant's y-component, and (in 3D) all the z-components for the resultant's z-component — each axis is summed completely independently of the others.

Enter dimension.
Enter number of vectors.
Enter vector A — x.
Enter vector A — y.
Enter vector A — z (3D only).
Enter vector B — x.
Enter vector B — y.
Enter vector B — z (3D only).
Enter vector C — x (if 3 vectors).
Enter vector C — y (if 3 vectors).
Enter vector C — z (3D only, if 3 vectors).
Click Calculate to see your results.

Examples

(1,2) + (3,4)

Sum = (1+3, 2+4) = (4, 6).

(1,0,-1) + (2,3,1) + (-1,1,2) in 3D

x: 1+2-1=2. y: 0+3+1=4. z: -1+1+2=2. Sum = (2, 4, 2).

Common mistakes to avoid

Leaving the third vector's fields non-zero when only 2 vectors are selected.
Adding magnitudes of the vectors directly instead of adding their components — this only works correctly when vectors point in exactly the same direction.
Mixing up which vector's x-component pairs with which — always sum same-axis components together.

Frequently asked questions

No — vector addition is commutative, so (A+B) always equals (B+A), and this extends to any number of vectors: you can add them in any order and get the same resultant.

If you draw the first vector, then start the second vector's tail at the first vector's head (and so on for more vectors), the resultant is the single arrow drawn from the very first tail to the very last head — exactly matching the component-wise sum.

Yes — if the vectors are arranged so they exactly cancel out (like two equal-magnitude, opposite-direction forces), their sum is the zero vector, representing equilibrium.

Adding the magnitudes (lengths) of two vectors ignores direction and gives a single number; vector addition preserves both magnitude and direction, and the resultant's length can be anywhere from the difference to the sum of the two magnitudes depending on the angle between them.

Yes — grouping doesn't matter either, so (A+B)+C always gives the same resultant as A+(B+C), which is why this calculator can support 2 or 3 vectors and still get a well-defined answer regardless of the order they're combined in.

Yes — adding a vector to itself scales every component by 2, which is equivalent to scalar multiplication by 2, producing a vector twice as long pointing in exactly the same direction.