Velocity & Position Vectors

Speed with a direction, built from the arrow that tracks where you are.

10 minUpdated 21 October 2026position vectoraverage velocityinstantaneous velocity

Lesson at a glanceShow the key points

The position vector points from the origin of the frame to the particle.
Average velocity = (r₂ − r₁)/(t₂ − t₁); it depends only on the start and end points.
Instantaneous velocity v = dr/dt is tangent to the path; its magnitude equals the speed.
Average velocity can be zero while average speed is not (any round trip).

Speed tells you how fast. Velocity tells you how fast and which way, and it is built on displacement, not distance.

Average velocity

If a particle is at A at time $t_1$ and at B at time $t_2$ , its displacement over that interval is the vector $\vec{AB}$ . Divide by the elapsed time:

Average velocity

\vec v_{av} = \frac{\vec{AB}}{t_2 - t_1}

Dividing a vector by a positive number keeps its direction, so average velocity points from A to B. It is a vector.

Definition · Position vector

The arrow drawn from the origin of the frame to the particle’s current location. If it is

\vec r_1

at time

t_1

and

\vec r_2

t_2

, then by the triangle rule the displacement is

\vec{AB} = \vec{AO} + \vec{OB} = \vec r_2 - \vec r_1

Average velocity, using position vectors(3.4)

\vec v_{av} = \frac{\vec r_2 - \vec r_1}{t_2 - t_1}

Interactive

Where is the tip of the minute hand going?

Start at 3:00. As time passes the tip piles up distance along its circle, but its displacement (and so its average velocity) depends only on where it ends up.

clock reads3:30

distance18.8cm

displacement12.00cm

avg speed1.05e-2cm/s

|avg velocity|6.67e-3cm/s

Minutes after 3:0030min

Worked example

The minute hand

A wall clock’s minute hand is 6.0 cm long. Find the average velocity of its tip (a) from 3:00 to 3:30 the same afternoon, and (b) from 3:00 pm to 3:30 am the next morning.

In a vanishingly short interval the displacement points along the path itself, so velocity is always tangent to the trajectory. And over such a short stretch the chord and the arc are the same length, $|d\vec r| = ds$ , so

|\vec v| = \left|\frac{d\vec r}{dt}\right| = \frac{ds}{dt} \qquad (3.6)

The magnitude of the velocity is the speed. Unlike the averages, which can be very different, the instantaneous speed and the size of the instantaneous velocity always agree.

Worked example

Laps of a running track

A runner completes 8 laps of a 400 m track in 32 minutes, finishing exactly where she began. Find her average speed and average velocity.

What to remember

1 $\vec v_{av} = (\vec r_2 - \vec r_1)/(t_2 - t_1)$ : depends only on the endpoints.
2 $\vec v = d\vec r/dt$ points along the tangent to the path.
3|instantaneous velocity| = instantaneous speed, but average velocity and average speed can differ wildly.

Check yourself

A car drives round a roundabout at a steady 30 km/h. Its velocity is…

Can a particle have zero average velocity but non-zero average speed over the same interval?

Keep exploring

Hand-picked videos, simulations and readings that explain this idea from a different angle.

WatchUnits, dimensions, uncertaintiesWalter Lewin · MIT 8.01x Lecture 1The famous ‘any measurement without uncertainty is meaningless’ lecture ReadPosition, displacement & average velocityOpenStax University Physics §3.1 ReadInstantaneous velocity and speedOpenStax University Physics §3.2 ReadDisplacement and velocity vectorsOpenStax University Physics §4.1

PreviousSpeed: Average & Instantaneous Next Acceleration

Velocity & Position Vectors

Speed with a direction, built from the arrow that tracks where you are.

10 minUpdated 21 October 2026position vectoraverage velocityinstantaneous velocity

Lesson at a glanceShow the key points

The position vector points from the origin of the frame to the particle.
Average velocity = (r₂ − r₁)/(t₂ − t₁); it depends only on the start and end points.
Instantaneous velocity v = dr/dt is tangent to the path; its magnitude equals the speed.
Average velocity can be zero while average speed is not (any round trip).

Speed tells you how fast. Velocity tells you how fast and which way, and it is built on displacement, not distance.

Average velocity

If a particle is at A at time $t_1$ and at B at time $t_2$ , its displacement over that interval is the vector $\vec{AB}$ . Divide by the elapsed time:

Average velocity

\vec v_{av} = \frac{\vec{AB}}{t_2 - t_1}

Dividing a vector by a positive number keeps its direction, so average velocity points from A to B. It is a vector.

Definition · Position vector

The arrow drawn from the origin of the frame to the particle’s current location. If it is

\vec r_1

at time

t_1

and

\vec r_2

t_2

, then by the triangle rule the displacement is

\vec{AB} = \vec{AO} + \vec{OB} = \vec r_2 - \vec r_1

Average velocity, using position vectors(3.4)

\vec v_{av} = \frac{\vec r_2 - \vec r_1}{t_2 - t_1}

Interactive

Where is the tip of the minute hand going?

Start at 3:00. As time passes the tip piles up distance along its circle, but its displacement (and so its average velocity) depends only on where it ends up.

clock reads3:30

distance18.8cm

displacement12.00cm

avg speed1.05e-2cm/s

|avg velocity|6.67e-3cm/s

Minutes after 3:0030min

Worked example

The minute hand

A wall clock’s minute hand is 6.0 cm long. Find the average velocity of its tip (a) from 3:00 to 3:30 the same afternoon, and (b) from 3:00 pm to 3:30 am the next morning.

Instantaneous velocity

As with speed, we zoom in. Take the displacement $\Delta\vec r$ in a short interval $\Delta t$ after time $t$ and let the interval shrink:

Velocity(3.5)

\vec v = \lim_{\Delta t\to 0}\frac{\Delta\vec r}{\Delta t} = \frac{d\vec r}{dt}

Interactive

Position vectors and the velocity arrow

O is the origin of the frame. The arrows from O locate the particle at two times; Δr joins their tips. Shrink Δt and Δr/Δt turns into the velocity, tangent to the path.

|\Delta\vec r|

2.40m

|\vec v_{av}|

0.80m/s

|\vec v|

1.19m/s

Interval

\Delta t

3.00s

|\vec v| = \left|\frac{d\vec r}{dt}\right| = \frac{ds}{dt} \qquad (3.6)

The magnitude of the velocity is the speed. Unlike the averages, which can be very different, the instantaneous speed and the size of the instantaneous velocity always agree.

Worked example

Laps of a running track

A runner completes 8 laps of a 400 m track in 32 minutes, finishing exactly where she began. Find her average speed and average velocity.

What to remember

1 $\vec v_{av} = (\vec r_2 - \vec r_1)/(t_2 - t_1)$ : depends only on the endpoints.
2 $\vec v = d\vec r/dt$ points along the tangent to the path.
3|instantaneous velocity| = instantaneous speed, but average velocity and average speed can differ wildly.

Check yourself

A car drives round a roundabout at a steady 30 km/h. Its velocity is…

Can a particle have zero average velocity but non-zero average speed over the same interval?

Keep exploring

Hand-picked videos, simulations and readings that explain this idea from a different angle.

PreviousSpeed: Average & Instantaneous Next Acceleration

Average velocity#

Instantaneous velocity#

Check yourself

Keep exploring

Average velocity#

Instantaneous velocity#

Check yourself

Keep exploring

Average velocity

Instantaneous velocity

Average velocity

Instantaneous velocity