Vector Addition Calculator

How it works

1. 1Choose 2D or 3D mode, then enter the x, y (and z) components for vectors a and b.
2. 2The calculator adds each pair of matching components — xₐ + x_b, yₐ + y_b, and zₐ + z_b — to form the resultant vector.
3. 3The magnitude |a + b| is computed as √(x² + y² + z²) of the resultant, giving the vector’s length in the same units as the inputs.

Use cases

• Physics problems that require combining force, velocity, or displacement vectors acting on an object.
• Computer graphics and game development where positions or motion directions are accumulated across frames.
• Engineering statics and dynamics courses when resolving multiple concurrent vectors into a single resultant.

Frequently asked questions

What is vector addition?

Vector addition combines two vectors by summing their corresponding components. For vectors a = (a₁, a₂, a₃) and b = (b₁, b₂, b₃) the result is (a₁ + b₁, a₂ + b₂, a₃ + b₃). The operation follows the parallelogram law geometrically.

What does the magnitude of the resultant represent?

The magnitude |a + b| is the Euclidean length of the resultant vector — the straight-line distance from the origin to the tip of the combined vector. It equals √(x² + y² + z²) and carries the same units as the input components.

Is vector addition commutative?

Yes. a + b = b + a for any two vectors. The order of entry does not affect the resultant sum or its magnitude.