Choose north as positive y and east as positive x, then record each movement as a coordinate change.

Turning left or right depends on the current facing direction, not the original direction throughout the entire journey.

A sketch can reduce memory load, but every turn and distance must be represented consistently.

Move 3 km east and then 4 km north from the origin. The final coordinates are (3,4) under the chosen convention.

Travelled distance is 7 km; shortest straight-line distance from the start is √(3²+4²)=5 km.

The final position is north-east of the origin. The distance travelled is not the same as displacement magnitude.

For multi-stage routes, total east–west and north–south components separately before calculating the straight-line distance.

A person who returns to the starting point has zero final displacement despite possibly travelling a positive distance.

Distinguish the direction the person faces from the direction of their final location relative to the starting point.

Choose a fixed coordinate convention: east is positive x, north is positive y, west is negative x, and south is negative y. Record every movement as a displacement and add the components. The origin is simply the starting point; it does not need a geographical latitude. Keeping the convention fixed prevents mistakes when a person changes the direction they face.

Worked example: a person travels 6 km east, 4 km north, 3 km west, and 2 km south. The final displacement is (6−3,4−2)=(3,2). The straight-line distance from the start is √(3²+2²)=√13 km, and the person is northeast of the origin. The total route length is 6+4+3+2=15 km. These are three different answers: displacement vector, shortest distance, and distance travelled.

For turns, update orientation separately from position. Facing north and turning right means facing east; facing south and turning right means west. Right is relative to the person’s current orientation, not permanently east. A small arrow for facing direction beside the coordinate table is often enough to keep these two kinds of information separate.

If A is 4 km west of B, then with B at (0,0), A is (−4,0). Reversing the relation, B is 4 km east of A. If C is 3 km north of A, C is (−4,3), and the shortest distance BC is 5 km. The 3–4–5 triangle provides a useful check, but the coordinate method works even without a familiar triple.

Do not use the Pythagorean theorem unless the components form a right-angle decomposition. A diagonal direction described as northeast may require checking whether the distance along the diagonal is given or whether separate north and east movements are specified. Two perpendicular legs of length 5 give a diagonal 5√2, not 10; a diagonal of 5 does not give both legs equal to 5.

When interpreting a diagram, distinguish what is stated from what merely looks equal or parallel. A rough sketch supports reasoning but is not an extra source of measured information. Check the final answer by reversing the displacement back to the start and by estimating whether the shortest distance could exceed the route travelled; it ordinarily cannot.