Definition

Vector addition

C = A + B

Statement

Two vectors combine into a third by adding corresponding components; the result's magnitude is not the sum of the magnitudes unless the vectors are parallel.

Symbols and units

$A, B$	vectors being added	—
$C$	resultant (sum) vector	—
$C_{x}, C_{y}$	components of the sum	—

When it applies

Any vectors of the same kind (e.g. two forces, not a force and a velocity).

Derivation

Geometrically, place the tail of B at the head of A; the sum runs from the tail of A to the head of B (the triangle rule).

Algebraically, add corresponding components:

C_{x} = A_{x} + B_{x}, C_{y} = A_{y} + B_{y}

Worked examples

Adding two displacements

A = (3, 4), B = (1, −2):

C = A + B = (4, 2)

Compare magnitudes:

∣ A ∣ = 5, ∣ B ∣ = 5 \approx 2.24, ∣ C ∣ = 20 \approx 4.47

4.47 is less than 5 + 2.24 = 7.24 — the magnitudes do not add unless the vectors are parallel.

Watch out for

|A + B| ≤ |A| + |B|, with equality only when A and B point the same way — magnitudes do not simply add.