Position

Level 1 - Math II (Physics) topic page in Kinematics.

Principle

Position describes where a particle is relative to a chosen origin and coordinate axes. In three-dimensional Cartesian coordinates, the position at time \(t\) is a vector \(\mathbf r(t)\) from the origin to the particle.

Displacement compares two positions. It is the vector change in position between two times, so it records both distance and direction of the change.

Notation

\(O\)

chosen origin of the coordinate system

\(t\)

time

\(\mathrm{s}\)

\(\mathbf r(t)\)

position vector of the particle at time t

\(\mathrm{m}\)

\(x(t), y(t), z(t)\)

Cartesian coordinate functions of position

\(\mathrm{m}\)

\(\mathbf i, \mathbf j, \mathbf k\)

unit vectors along the x, y, and z axes

\(\Delta\mathbf r\)

displacement between two positions

\(\mathrm{m}\)

Method

Step 1: Choose the reference frame

State the origin \(O\), axes, and units. A position vector depends on this choice, because changing the origin changes the vector from the origin to the particle.

Step 2: Write the position vector

In Cartesian coordinates, write the position as components along the basis vectors:

Position vector

\[\mathbf r(t)=x(t)\mathbf i+y(t)\mathbf j+z(t)\mathbf k\]

Step 3: Compare two positions

For positions at times \(t_1\) and \(t_2\), subtract final minus initial:

Displacement

\[\Delta\mathbf r=\mathbf r(t_2)-\mathbf r(t_1)\]

The displacement uses the same unit as position: metres.

Rules

Component displacement

\[\Delta\mathbf r=(x_2-x_1)\mathbf i+(y_2-y_1)\mathbf j+(z_2-z_1)\mathbf k\]

Distance from origin

\[|\mathbf r|=\sqrt{x^2+y^2+z^2}\]

Position depends on the chosen origin.
Displacement is final position minus initial position.
Position and displacement are vectors, so direction matters.
The length \(|\Delta\mathbf r|\) is the straight-line distance between the two positions, not necessarily the distance travelled along a curved path.

Examples

Question

A particle has position

\[\mathbf r=3\mathbf i-2\mathbf j+5\mathbf k\]

m. What are its coordinates?

Answer

Read the coefficients of the Cartesian basis vectors. The coordinates are

\[(3,-2,5)\]

m relative to the chosen origin.

Checks

Do not confuse position with displacement. Position locates one event; displacement compares two events.
Always state or infer the origin and axes before interpreting a position vector.
Keep the unit of position and displacement as metres unless the model uses a different length unit.
The displacement vector can be zero even when the particle has travelled along a path and returned to its starting point.

Planes

Velocity