Distance, Displacement, Vectors, and Scalars

Is "how far you drove" the same kind of fact as "how far from home you ended up"? They're both measured in miles — but in the 1880s, that question actually caused a real, public fight between a physicist and an engineer.

Josiah Willard Gibbs and Oliver Heaviside were working independently, on opposite sides of the Atlantic, trying to simplify the math behind James Clerk Maxwell's new equations for electricity and magnetism. The standard tool at the time was quaternions, invented decades earlier by William Rowan Hamilton — a single kind of number that permanently fused together a "size-only" part and a "size-and-direction" part. Gibbs and Heaviside both independently decided that was the wrong move, and split it apart: quantities with direction, and quantities that are just a size, kept as two genuinely different kinds of things. It wasn't a quiet disagreement — Peter Tait, one of Hamilton's own former students, publicly called Gibbs's version a "hermaphrodite monster" in print, and Heaviside fired back in the journal Nature. Gibbs and Heaviside won, not by being louder, but because the split turned out to be easier to actually use — and it's the exact split every physics class, and every video-game physics engine, still runs on today.

That's the split behind this whole lesson. If you drive to the grocery store and back home, your car's odometer racks up miles the whole way — but by the time you pull back into your driveway, you haven't actually gone anywhere. That contradiction is the starting point for how physicists describe motion.

Distance is the total length of the path you actually traveled, no matter which way you turned along the way. It's always a positive number, and it doesn't care about direction — it's what your odometer measures.

Displacement is different. It only cares about your starting point and your ending point, measured as a straight line between the two. If you end up back where you started, your displacement is zero — even if you drove twenty miles to get there. Displacement is written with the symbol Δx (the Greek letter delta means "change in," so Δx means "change in position").

That difference — caring about direction or not — turns out to be a huge deal in physics, so it gets its own category. A scalar is any quantity that's just a number: distance, speed, time, mass. A vector is a quantity that needs a direction to be complete: displacement, velocity, acceleration, force. Saying "I walked 10 miles" is a distance (scalar). Saying "I walked 10 miles north" is a displacement (vector) — and leaving off the "north" makes the answer incomplete, even if the number is right.

Vectors that point along the same line are easy to combine: walk 5 meters north, then 3 more meters north, and you've gone 8 meters north (same direction, so you add). Walk 5 meters north, then 3 meters south, and you've only ended up 2 meters north of where you started (opposite directions, so you subtract). This simple idea — add when directions match, subtract when they don't — is the foundation for a huge amount of physics that follows.

Try It: Treasure Trek

Walk a path and compare distance with displacement. Kinematics Castle: Treasure Trek

Videos

• Distance vs Displacement — YouTube
• What Are Vectors and Scalars? — YouTube

Practice Problems

1. A jogger runs 12 m east, then continues 5 m further east. Find the total distance traveled and the displacement.
2. Classify each as a vector or a scalar: mass, velocity, time, force.
3. A cyclist rides 400 m east to a store, then rides 250 m back west before stopping. Find the total distance traveled and the displacement (magnitude and direction).
4. An elevator goes up 3 floors, then up 2 more, then down 4. Find the net floor change, and state whether "net floor change" here is being used as a vector or scalar quantity.
5. A hiker walks 3 km north, then 2 km east, then 3 km south, ending the hike. Find the total distance traveled and the displacement (magnitude and direction).
6. A classmate claims: "Whenever an object returns exactly to its starting point, both its distance traveled and its displacement are zero." Create your own numerical example to test this claim, calculate both quantities, and explain whether the claim is true or false.
7. A small robot moves along a straight track: 5 m east, then 3 m further east, then 2 m west. Find the robot's total distance traveled and total displacement, and explain whether the robot's direction of motion changed at any point.
8. A cyclist rides a winding trail through a park and returns to the exact spot where they started. Which method would give the most accurate measurement of the cyclist's displacement for the entire ride: (A) summing the odometer reading for every segment of the trail, (B) recording GPS coordinates at the start and end and calculating the straight-line distance between them, (C) multiplying average speed by total time, or (D) counting the number of turns taken?