# Definition:Position Vector

## Definition

Let $P$ be a point in a given frame of reference whose origin is $O$.

The position vector $\mathbf p$ of $P$ is the displacement vector of $P$ from $O$.

### Notation

When considering a position vector $\mathbf r$ with respect to the origin $O$ of a point $P$ in space under a Cartesian coordinate system, it is commonplace to refer to it as:

$P = \tuple {x, y, z}$

where $x$, $y$ and $z$ are the components of $\mathbf r$ in the directions of the coordinate axes.

Hence $P = \tuple {x, y, z}$ can be regarded as shorthand for:

$\mathbf r = x \mathbf i + y \mathbf j + z \mathbf k$

where $\mathbf i$, $\mathbf j$ and $\mathbf k$ are unit vectors along the $x$-axis, $y$-axis and $z$-axis from $O$ respectively.

## Also known as

The position vector of a point $P$ is also known, particularly with respect to a polar coordinate system or a spherical coordinate system, as its radius vector.

## Also see

• Results about position vectors can be found here.