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Vector Addition — Definition, Formula & Examples

Vector addition is the operation of combining two or more vectors to produce a new vector called the resultant. You add the corresponding components of each vector together.

Given vectors u=⟨u1,u2,…,un⟩\mathbf{u} = \langle u_1, u_2, \ldots, u_n \rangle and v=⟨v1,v2,…,vn⟩\mathbf{v} = \langle v_1, v_2, \ldots, v_n \rangle in Rn\mathbb{R}^n, their sum is defined as u+v=⟨u1+v1,u2+v2,…,un+vn⟩\mathbf{u} + \mathbf{v} = \langle u_1 + v_1, u_2 + v_2, \ldots, u_n + v_n \rangle. Vector addition is commutative and associative.

Key Formula

u+v=⟨u1+v1,  u2+v2⟩\mathbf{u} + \mathbf{v} = \langle u_1 + v_1,\; u_2 + v_2 \rangle
Where:
  • u\mathbf{u} = First vector with components $u_1$ and $u_2$
  • v\mathbf{v} = Second vector with components $v_1$ and $v_2$
  • u+v\mathbf{u} + \mathbf{v} = Resultant vector

How It Works

To add two vectors, pair up the components in matching positions and add each pair. For 2D vectors, add the xx-components together and the yy-components together. Geometrically, place the tail of the second vector at the tip of the first — the resultant vector goes from the start of the first to the tip of the second (the "tip-to-tail" method). This works identically in three or more dimensions.

Worked Example

Problem: Find the sum of u=⟨3,−1⟩\mathbf{u} = \langle 3, -1 \rangle and v=⟨−5,4⟩\mathbf{v} = \langle -5, 4 \rangle.
Add x-components: Add the first components of each vector.
3+(−5)=−23 + (-5) = -2
Add y-components: Add the second components of each vector.
−1+4=3-1 + 4 = 3
Write the resultant: Combine the results into a single vector.
u+v=⟨−2,3⟩\mathbf{u} + \mathbf{v} = \langle -2, 3 \rangle
Answer: u+v=⟨−2,3⟩\mathbf{u} + \mathbf{v} = \langle -2, 3 \rangle

Why It Matters

Vector addition is fundamental in physics for combining forces, velocities, and displacements. In precalculus and linear algebra courses, it serves as the building block for nearly every other vector operation you will encounter.

Common Mistakes

Mistake: Adding the magnitudes of two vectors instead of their components.
Correction: The magnitude of the sum is generally not equal to the sum of the magnitudes. Always add component by component: ∥u+v∥≠∥u∥+∥v∥\|\mathbf{u} + \mathbf{v}\| \neq \|\mathbf{u}\| + \|\mathbf{v}\| in most cases.