Position

In order to describe the motion of an object, you must first be able to describe its position (where it is at any particular time). More precisely, you need to specify its position relative to a convenient reference frame.

So the position of a point \(P\) can be described by a pair or a set of coordinates, such as: \(P=(x, y)\) (in two dimensions) or \(P=(x, y, z)\) (in three dimensions).

Position vector

The position vector, also known as location vector or radius vector, is a Euclidean vector that represents the position of a point \(P\) in space in relation to an arbitrary reference origin \(O\). Usually denoted \(\vec{r}\), or \(\vec{s}\), it corresponds to the straight-line from \(O\) to \(P\).

In other words, it is the displacement or translation that maps the origin to \(P\). In one dimension \(x(t)\) is used to represent position as a function of time. In two dimensions, either cartesian or polar coordinates may be used, and the use of unit vectors is common. A position vector \(r\) may be expressed in terms of the unit vectors as follow:

\[\vec{r}(t)=x\vec{i}+y\vec{j}\]

In three dimensions, cartesian or spherical polar coordinates are used, as well as other coordinate systems for specific geometries.

\[\vec{r}(t)=x\vec{i}+y\vec{j}+z\vec{k}\]

The vector change in position associated with a motion is called the displacement.

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