Magnitude of a 3D Vector

|\vec{v}| = \sqrt{v_x^2 + v_y^2 + v_z^2}

Worked example: Components (2, 3, 6) → magnitude 7 — press Try an example to run it live, then adjust anything.

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Magnitude of a 3D Vector explained

In space the same rule simply picks up a third term: apply Pythagoras once in the xy-plane, then again with that diagonal and the z-leg, and the two steps collapse into √(v_x² + v_y² + v_z²). It is the length of the space diagonal of a box with sides v_x, v_y, v_z. Every phone in a pocket computes it thousands of times a second: a three-axis accelerometer at rest reports something like (0.20, −1.10, 9.72) m/s², and the magnitude √(0.04 + 1.21 + 94.48) ≈ 9.79 m/s² is gravity, whatever way the handset happens to be tilted.

The components (2, 3, 6) are a favourite of textbook writers because 4 + 9 + 36 = 49 lands exactly on 7. Solving backwards for a missing component takes the positive root and requires the other two to leave something under the radical: if v_x² + v_y² already exceeds |v|², the vector you described cannot be built.

Magnitude of a 3D Vector formula

|\vec{v}| = \sqrt{v_x^2 + v_y^2 + v_z^2}

Where

$|\vec{v}|$ = Vector magnitude
$v_x$ = x-component
$v_y$ = y-component
$v_z$ = z-component

Missing one of these? Work it out first, then come back

Vector magnitude — Magnitude of a 2D Vector, Unit Vector Component (Normalization)
x-component — Magnitude of a 2D Vector, Dot Product of Two 2D Vectors (Components)
y-component — Magnitude of a 2D Vector, Dot Product of Two 2D Vectors (Components)
z-component — Magnitude of a 2D Vector, Dot Product of Two 2D Vectors (Components)

Part of this formula set

Vector operations

12 formulasMagnitude, direction, components and unit vectors, plus the dot and cross products and the projections and angles they give you.