Definition

Components and magnitude

A = A_{x}  + A_{y} , ∣ A ∣ = A_{x}^{2} + A_{y}^{2}

Statement

A vector is written as a sum of axis components; its magnitude follows from the Pythagorean theorem.

Symbols and units

When it applies

Any Cartesian coordinate system; the axes must be mutually perpendicular.

Derivation

Project the vector onto each axis to obtain its components:

A_{x} = ∣ A ∣ cos θ, A_{y} = ∣ A ∣ sin θ

The magnitude is recovered from the components:

∣ A ∣ = A_{x}^{2} + A_{y}^{2}

Worked examples

Magnitude and quadrant from components

A_x = −3, A_y = 4:

∣ A ∣ = (- 3)^{2} + 4^{2} = 5

The reference angle from arctan alone gives 53.1°, but the negative x-component places A in the second quadrant:

θ = 18 0^{\circ} - 53. 1^{\circ} = 126. 9^{\circ}

Watch out for

The angle from arctan(A_y/A_x) alone does not fix the quadrant — check the signs of A_x and A_y separately.