Position

Concept & Definition

To describe how an object moves, the first question we must answer is straightforward: where is it located? In mechanics, position specifies the location of a particle in space relative to an agreed-upon reference point, known as the origin ($O$).

The position vector, denoted by $\vec{r}$ (or bold $\mathbf{r}$), is a directed arrow drawn from the origin directly to the point where the particle is situated. Because specifying a location requires both how far away the particle is and in which direction it lies, position is fundamentally a vector quantity.

Units and Dimensions

The SI unit of position is the meter ($\text{m}$), and its dimensional formula is $[\text{L}]$.

Coordinate Systems and Dimensions

Position can be defined across different spatial dimensions depending on the constraints of motion:

• In one dimension (1D), motion is confined to a straight line (such as the $x$-axis). The position is specified by a single signed coordinate $x$, where $\vec{r} = x\hat{i}$.
• In two dimensions (2D), motion occurs in a plane. The position is specified by an ordered pair $(x, y)$, where $\vec{r} = x\hat{i} + y\hat{j}$.

Figure 1: Position vector $\vec{r}$ along a 1D axis from origin $O$ to point $P(x)$.

Figure 2: Position vector $\vec{r}$ drawn from origin $O$ to point $P(x, y)$.

In three dimensions (3D), the particle moves freely through physical space, requiring three mutually perpendicular coordinates $(x, y, z)$ to determine its exact location.

Figure 3: Position vector $\vec{r}$ of point $P(x, y, z)$ in a 3D Cartesian coordinate system.

Mathematical Formulation

In a three-dimensional Cartesian coordinate system with origin $O(0, 0, 0)$, the position vector of any point $P(x, y, z)$ is expressed in terms of the standard unit vectors:

Key Formula

Position Vector in 3D

$$\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$$

where $(x, y, z)$ are the Cartesian coordinates of the particle and $(\hat{i}, \hat{j}, \hat{k})$ are unit vectors pointing along the $x$, $y$, and $z$ axes.

Magnitude and Distance

The magnitude of the position vector represents the straight-line distance from the origin $O$ to the particle at point $P$. By applying the three-dimensional Pythagorean theorem, this distance is:

Key Formula

Magnitude from Origin

$$r = |\vec{r}| = \sqrt{\rule{0pt}{2.7ex}x^2 + y^2 + z^2}$$

where $r = |\vec{r}|$ is the scalar distance from the origin $(0, 0, 0)$ to the particle at $(x, y, z)$.

Relative Position and Reference Frames

Position has no absolute meaning on its own—it is always defined relative to a chosen frame of reference. If the observer is not at the origin $(0, 0, 0)$ but located at coordinates $\vec{r}_0 = (x_0, y_0, z_0)$, the position vector of the particle relative to the observer is given by:

Key Formula

Relative Magnitude

$$|\vec{r} - \vec{r}_0| = \sqrt{\rule{0pt}{2.7ex}(x - x_0)^2 + (y - y_0)^2 + (z - z_0)^2}$$

where $(x, y, z)$ are the coordinates of the particle and $(x_0, y_0, z_0)$ are the coordinates of the observer.

Reference Frame Dependency

Position is not an intrinsic property of a particle. Changing the origin or rotating the coordinate axes changes the coordinate values and the position vector $\vec{r}$, even if the physical object remains completely stationary relative to the surroundings.