Distinguish a displacement vector from a route

1. Orient: Distinguish a displacement vector from a route

A displacement connects initial and final positions. Distance records the length of the route actually traveled. On a straight axis, opposite displacement components carry opposite signs; adding distances instead uses nonnegative lengths. A round trip can have zero displacement and nonzero distance. A scalar has magnitude without direction: distance and speed are examples. A vector has magnitude and direction: displacement, velocity and acceleration are examples. An arrow above a symbol denotes the whole vector; a signed x or y component does not need that arrow because the axis and sign specify its direction.

2. Explain the model

For perpendicular east and north axes, represent a displacement by the ordered pair (Δx,Δy). Its magnitude is √(Δx²+Δy²), from the right triangle joining the endpoints. This is not generally |Δx|+|Δy|, which is the length of a particular two-leg route. A diagonal route and a right-angle route can share a displacement but differ in distance.

A direction needs both the signs of the components and a reference axis. The angle whose tangent is Δy/Δx is ambiguous without a quadrant: northwest and southeast can give the same ratio. The visualization uses the quadrant-aware angle from east, positive counterclockwise. The zero vector has no unique direction.

Components are not extra vectors secretly added to the motion. They are coordinate descriptions of one vector. Rotating the axes changes the component values but not the vector magnitude. Without an elapsed time, neither the displacement nor the route determines an average velocity or average speed. If a vector has magnitude R and angle θ measured counterclockwise from east, its components are R cos θ and R sin θ. These expressions follow from adjacent/hypotenuse and opposite/hypotenuse in the right triangle, with signs determined by the quadrant. For R=10 m and θ=30°, the components are about 8.66 m east and 5 m north. Add a second vector by adding its east component to the first east component and likewise for north; only then compute the resultant magnitude. A 5 m westward vector added to this example gives (3.66,5) m, not a 5 m resultant.

3. Work an example

Moving 3 m east then 4 m north gives displacement (3,4) m, magnitude √(9+16)=5 m and route distance 7 m. Moving 3 m west then 4 m north still gives magnitude 5 m but changes the direction. The arrow length alone cannot encode that difference.

4. Predict and investigate

Predict first: If one leg reverses while its length stays fixed, which of route distance, displacement magnitude and direction changes?

Compare (3,4), (−3,4) and (0,4). Sketch each route and direct displacement. Find when the route distance equals the displacement magnitude and explain the equality geometrically. Test the zero-vector case rather than assigning it a physical angle. Use the lesson experiment controls and retain the settings with your evidence.

5. Transfer and explain

Try a new practice case

A robot moves 5 m west and then 12 m north. Find its displacement magnitude, not its route distance.

Your numerical result (m) Check number

Reveal worked check (practice, not mastery)

This extra practice is not saved and does not award mastery. Record your first answer, explanation and whether you revealed help in the lesson notebook. Use the lesson’s saved question separately.

6. Reflect and retain evidence

Give two routes with the same endpoints but different distances. What clock evidence would be needed to compare their speeds?

Record a claim, numerical or graphical evidence, assumptions, a limitation and a changed-case prediction. A supplied synthetic investigation is not evidence that you performed a physical laboratory.

Framework and source notes

Original teaching prose, cases and synthetic datasets. Framework mapping: 1.1.A; 1.1.B; 1.5.A. College Board AP Physics 1 CED, Unit 1, checked 30 October 2026. This mapping is a content cross-reference, not AP authorization or a claim of full course completion.