Vector Addition Calculator

Vector Addition in 2D and 3D

Vector addition combines displacements by adding matching coordinates. If $\mathbf{a} = (a_x, a_y, a_z)$ and $\mathbf{b} = (b_x, b_y, b_z)$, then:

$$\mathbf{a} + \mathbf{b} = (a_x + b_x,\ a_y + b_y,\ a_z + b_z)$$

This calculator also supports scalar multiples: $\alpha\mathbf{a} + \beta\mathbf{b}$. See also the Vector Calculator and Linear Combination Calculator.

Parallelogram Rule

Graphically, place the tail of $\mathbf{b}$ at the tip of $\mathbf{a}$. The diagonal from the origin to the new tip is $\mathbf{a} + \mathbf{b}$.

Frequently Asked Questions

How do I add vectors (2, 1) and (1, 0)?

Add each coordinate: (2 + 1, 1 + 0) = (3, 1).

Is vector addition commutative?

Yes. a + b = b + a because addition of real numbers is commutative.

How do I add three copies of a vector?

Set α = 1 and β = 3 to compute a + 3b, or add the vector to itself three times component-wise.

What is the magnitude of a + b?

|a + b| = √[(a_x + b_x)² + (a_y + b_y)² + (a_z + b_z)²]. Use the Vector Magnitude Calculator on the result.

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Vector Addition in 2D and 3D

Parallelogram Rule

Frequently Asked Questions

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