# Definition:Position Vector

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## 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 in some sources as its **radius vector**.

## Sources

- 1921: C.E. Weatherburn:
*Elementary Vector Analysis*... (previous) ... (next): Chapter $\text I$. Addition and Subtraction of Vectors. Centroids: Centroids: Definition - 1927: C.E. Weatherburn:
*Differential Geometry of Three Dimensions: Volume $\text { I }$*... (next): Introduction:Vector Notation and Formulae - 1951: B. Hague:
*An Introduction to Vector Analysis*(5th ed.) ... (previous) ... (next): Chapter $\text I$: Definitions. Elements of Vector Algebra: $2$. Graphical Representation of Vectors - 1970: George Arfken:
*Mathematical Methods for Physicists*(2nd ed.) ... (previous) ... (next): Chapter $1$ Vector Analysis $1.1$ Definitions, Elementary Approach