Position Vector

In physics, knowing where something is matters just as much as knowing how fast it moves. A position vector is a mathematical tool that describes the exact location of an object relative to a chosen reference point called the origin. Unlike a simple distance measurement, a position vector carries both magnitude (how far the object is from the origin) and direction (which way you must travel to reach it). This makes it a true vector quantity, and it forms the foundation of all kinematics — the study of motion. Think of a city map. If someone asks 'where is the library?', saying '3 kilometres away' is not enough — you also need to say '3 kilometres north-east'. The position vector does exactly this: it packages distance and direction into a single, precise mathematical object. In two dimensions it is written as a pair of components along the x- and y-axes; in three dimensions a third component along the z-axis is added. Every other kinematic quantity — displacement, velocity, acceleration — is ultimately built on top of position vectors. Position vectors are always measured from the origin of a chosen coordinate system. The choice of origin is arbitrary (you can place it wherever is convenient), but once chosen it must remain fixed for all measurements in that problem. Changing the origin shifts every position vector, so consistency is essential. The symbol used for a position vector is typically r (bold) or r with an arrow above it, and its SI unit is the metre (m).

Key takeaways
  • Choosing a coordinate system and origin
  • Writing the position vector in component form
  • Finding the magnitude of a position vector
  • Direction of a position vector
  • Position vectors vs displacement vectors

Choosing a coordinate system and origin

Before you can write down a position vector, you must set up a coordinate system. In 2D kinematics this means choosing two perpendicular axes (conventionally called x and y) and fixing an origin O where they cross. The origin might be the starting point of a journey, the centre of a room, or any other convenient reference. Everything else is measured relative to this fixed point.

Writing the position vector in component form

If an object P sits at a point that is x metres along the horizontal axis and y metres along the vertical axis, its position vector is written as r = xî + yĵ, where î and ĵ are unit vectors (length 1) pointing along the x- and y-axes respectively. In 3D, a third term zk̂ is added. The components x, y (and z) can be positive, negative, or zero — they tell you which side of the origin the object is on.

Finding the magnitude of a position vector

The magnitude |r| gives the straight-line distance from the origin to the object. By Pythagoras' theorem in 2D: |r| = √(x² + y²). In 3D this extends to |r| = √(x² + y² + z²). The magnitude is always a non-negative scalar — it tells you how far, while the full vector also tells you in which direction.

Direction of a position vector

The direction is usually quoted as an angle θ measured anticlockwise from the positive x-axis: θ = arctan(y/x). Be careful with the quadrant — if x is negative, you must add 180° to the arctan result to get the correct angle. Alternatively, you can simply leave the vector in component form, which implicitly contains all directional information.

Position vectors vs displacement vectors

A position vector r points from the origin to the object. A displacement vector Δr points from one position to another and equals the difference of two position vectors: Δr = r₂ − r₁. This distinction is crucial: two objects can have the same displacement but completely different position vectors if they started in different places.

The Key Equation

r⃗=x i^+y j^+z k^\vec{r} = x\,\hat{i} + y\,\hat{j} + z\,\hat{k}
  • \vec{r} — Position vector (m): The vector from the origin O to the object's location; fully describes where the object is.
  • x — x-component (m): Signed distance of the object from the origin along the horizontal (x) axis.
  • y — y-component (m): Signed distance of the object from the origin along the vertical (y) axis.
  • z — z-component (m): Signed distance of the object from the origin along the depth (z) axis; omitted in 2D problems.
  • \hat{i},\,\hat{j},\,\hat{k} — Unit vectors (dimensionless): Vectors of magnitude 1 pointing along the x-, y-, and z-axes respectively; they define the directions of the coordinate system.

Worked Example

Problem. A drone starts from a launch pad taken as the origin. It flies to a position 40 m east and 30 m north of the pad. (a) Write the position vector of the drone. (b) Calculate the magnitude of the position vector. (c) Find the angle the position vector makes with the east direction.

r⃗=40 i^+30 j^ m,∣r⃗∣=50 m,θ=36.9°\vec{r} = 40\,\hat{i} + 30\,\hat{j}\text{ m},\quad |\vec{r}| = 50\text{ m},\quad \theta = 36.9°

Answer. Position vector: 40î + 30ĵ m; magnitude 50 m; direction 36.9° north of east.

Figure
Diagram
xyOr⃗x = 40 my = 30 mP (drone)θPosition vector r⃗ = x î + y ĵ, with x and y components shown as dashed lines.
Origins · 1 minRené Descartes and how this was discovered
RD
René Descartes
1596–1650 · French

Descartes invented the Cartesian coordinate system — the grid of perpendicular axes that underpins all modern vector mathematics. By assigning numerical coordinates to every point in space, he made it possible to express geometric ideas as algebraic equations. His 1637 work La Géométrie laid the groundwork for analytic geometry, without which the concept of a position vector would have no mathematical home. Every time you write r = xî + yĵ, you are working in Descartes' framework. Legend has it that Descartes conceived the idea of coordinate geometry while lying in bed watching a fly crawl across the ceiling — he realised he could describe the fly's exact position using only two numbers: its distances from two adjacent walls.

  • Displacement Vector — Displacement is defined as the change in position vector (Δr = r₂ − r₁), so understanding position vectors is an essential prerequisite.
  • Vector Equality — Two position vectors are equal only if they have identical magnitude and direction — understanding vector equality clarifies when two objects occupy the same location.
  • Scalar and Vector Quantities — A position vector is a fundamental example of a vector quantity; reviewing scalars vs vectors explains why direction is essential to its definition.
  • Velocity Vector — Velocity is the rate of change of the position vector with respect to time, making position vectors the direct starting point for studying motion.