Displacement Vector
Displacement is one of the most fundamental ideas in kinematics. It describes the change in an object's position — not how far it has travelled, but how far it is from where it started, and in which direction. Because displacement has both a magnitude (size) and a direction, it is a vector quantity. This distinguishes it sharply from distance, which is a scalar and only records the total length of the path travelled, regardless of direction. Imagine walking 400 m around three sides of a square field and ending up 200 m due east of your starting point. Your distance travelled is 400 m, but your displacement is only 200 m east. The path you took is completely irrelevant to displacement — only the starting point and the finishing point matter. This path-independence is the defining feature of displacement and makes it enormously useful in physics, because equations of motion are built around displacement, not distance. Displacement vectors can be added, subtracted, and compared using the rules of vector algebra. Two displacement vectors are equal only if they share both the same magnitude and the same direction — a rule called vector equality. Understanding displacement is the gateway to velocity, acceleration, and the full machinery of classical mechanics.
- Position as a Vector
- Defining Displacement
- Path Independence
- Vector Equality
- Components and Magnitude
Position as a Vector
Before defining displacement, we need position. In one dimension, an object's position x is its location along a number line relative to a chosen origin. In two or three dimensions, position is described by a position vector r drawn from the origin to the object's location. The choice of origin is arbitrary, but once chosen it must stay fixed throughout a problem.
Defining Displacement
Displacement Δr (or Δx in 1-D) is the vector from the initial position to the final position: Δr = r_final − r_initial. The Greek letter Δ (delta) always means 'final minus initial'. Because it is a subtraction of vectors, the result is itself a vector with a specific magnitude and direction. The magnitude |Δr| is the straight-line distance between the two points — the shortest possible path between them.
Path Independence
No matter how winding or complex the actual journey, the displacement vector is always the straight arrow connecting start to finish. Think of it like a crow flying in a straight line between two towns — it doesn't follow the road. A car driving 5 km north, then 3 km east has travelled 8 km, but its displacement is √(5² + 3²) = √34 ≈ 5.83 km in the direction arctan(3/5) ≈ 31° east of north.
Vector Equality
Two vectors are equal if and only if they have the same magnitude AND the same direction. This means a displacement vector can be drawn anywhere on a diagram — its position on the page does not define it. For example, every displacement of '3 m due north' is identical, regardless of where the arrow is drawn. This property, called free vector equivalence, is what allows vectors to be moved around freely when performing addition using the tip-to-tail method.
Components and Magnitude
In two dimensions, a displacement vector Δr can be broken into perpendicular components: Δx (horizontal) and Δy (vertical). The magnitude is found using Pythagoras' theorem: |Δr| = √(Δx² + Δy²), and the direction angle θ satisfies tan θ = Δy / Δx. Components make it straightforward to add multiple displacements: simply add all the x-components together and all the y-components together, then find the resultant magnitude and direction.
The Key Equation
- \vec{\Delta r} — Displacement vector (metre (m)): The vector from the initial position to the final position; independent of the path taken.
- \vec{r}_{\!f} — Final position vector (metre (m)): The position vector of the object at the end of the motion, measured from the origin.
- \vec{r}_{\!i} — Initial position vector (metre (m)): The position vector of the object at the start of the motion, measured from the origin.
- \Delta x,\, \Delta y — Horizontal and vertical components (metre (m)): The projections of the displacement onto the x- and y-axes respectively.
- |\vec{\Delta r}| — Magnitude of displacement (metre (m)): The straight-line distance between initial and final positions; always less than or equal to total distance travelled.
Worked Example
Problem. A drone starts at position (2 m, 5 m) relative to a launch pad. It flies to a new position (8 m, 1 m). Find the displacement vector, its magnitude, and the angle it makes below the positive x-axis.
Answer. Displacement vector: (6 m, −4 m); magnitude ≈ 7.21 m at 33.7° below the positive x-axis.
Origins · 1 minRené Descartes and how this was discovered
Descartes invented the Cartesian coordinate system — the x–y grid that underpins the mathematical description of position and displacement. By assigning numerical coordinates to points in space, he made it possible to express geometric relationships as algebraic equations, directly enabling the component-based treatment of vectors that students use today when resolving displacement into Δx and Δy. Legend has it that Descartes conceived of the coordinate system while lying in bed, watching a fly crawl across his ceiling — he realised he could describe the fly's position precisely using two distances from the corner of the room.
Related Concepts
- Position and Reference Frames — Displacement is defined as the change in position, so understanding how position is measured relative to an origin and reference frame is an essential prerequisite.
- Distance vs Displacement — Distance and displacement are frequently confused; studying them together clarifies why displacement is path-independent while distance depends on the full route taken.
- Velocity Vector — Velocity is defined as displacement divided by time, so displacement is the foundational vector quantity from which velocity is derived.
- Vector Addition and Subtraction — Calculating net displacement from multiple successive displacements requires vector addition, making this a direct and immediate application of displacement concepts.