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.
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
- 1957: D.E. Rutherford: Vector Methods (9th ed.) ... (previous) ... (next): Chapter $\text I$: Vector Algebra: $\S 9$
- 1970: George Arfken: Mathematical Methods for Physicists (2nd ed.) ... (previous) ... (next): Chapter $1$ Vector Analysis $1.1$ Definitions, Elementary Approach
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): spherical coordinate system
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): spherical coordinate system