Magnitude of a Vector

Calculates the magnitude of the vector (x, y, z) as √(x² + y² + z²). For a plane vector, set z to 0.

Explanation

The magnitude of a vector, also called its length or norm, comes straight from the Pythagorean theorem: square each component, add them, take the square root.

|\vec{v}| = \sqrt{x^2 + y^2 + z^2}

$x$ , $y$ , $z$ — the components of the vector
$|\vec{v}|$ — the magnitude, the length of the arrow

For a vector in the plane, set $z$ to 0. The formula reduces to $\sqrt{x^2 + y^2}$ , which is exactly the distance from the origin to the point $(x, y)$ .

Example

Take $(x, y, z) = (3, 4, 0)$ .

|\vec{v}| = \sqrt{3^2 + 4^2 + 0^2} = \sqrt{9 + 16} = \sqrt{25} = 5

The magnitude is 5.

Notes

Components get squared, so a magnitude is never negative. The vector $(-3, -4, 0)$ also has magnitude 5.
The only vector with magnitude 0 is the zero vector, every component of which is 0.
Magnitude carries no direction. To isolate direction, divide each component by the magnitude to get the unit vector $\left( \dfrac{3}{5},\ \dfrac{4}{5},\ 0 \right)$ , whose magnitude is 1.
Scaling a vector by $k$ scales its magnitude by $|k|$ . A negative $k$ reverses the arrow but leaves the length positive.