Find the magnitude (length) and unit vector of a vector in any dimension. Enter the components separated by commas.
Usage Tip
Dividing a vector by its magnitude gives a unit vector — useful for describing pure direction.
unit vector = each component ÷ magnitude
The unit vector points the same way with length 1.
The unit vector and sum of squares are shown.
A zero vector has no defined direction.
The magnitude of a vector is its length, found by squaring each component, adding them, and taking the square root. This is the Pythagorean theorem extended to any number of dimensions. The unit vector divides each component by that length, giving a vector of length one that points the same way.
Unit vectors are useful whenever only a direction matters, such as describing the direction of a force or a surface normal. The zero vector has no direction, so it has no unit vector. Check that all components use the same units, since mixing units gives a magnitude with no physical meaning.