Cardinal Directions — Fundamental Concepts

Fundamental Concepts

Keep three records

Position: choose the start as (0, 0), east as positive x and north as positive y. Add signed movements along the relevant axis. North and east distances do not cancel each other.

Facing: maintain the clockwise cycle north → east → south → west. A right-angle right turn advances one step; a left turn moves one step backwards. Update facing before adding the next movement.

Distance: path length adds all travelled legs. Straight-line separation uses the final horizontal and vertical differences. A closed route can have positive path length and zero displacement.

A fresh coordinate check

A route goes 12 metres west, 5 metres south and 4 metres east. It ends at (-8, -5), southwest of the start. Path length is 21 metres; shortest separation is the square root of 89 metres. Neither 13 nor 21 is the straight-line answer.

For two people, place both in one coordinate system and subtract their positions. Changing the reference person reverses the direction but preserves the separation.

Conditions to preserve

Use consistent units.
Do not replace a quadrant such as northeast with an exact 45-degree bearing unless equal components or another condition establishes it.
At the starting point, zero displacement has no nonzero direction.
If a turn or distance is unspecified, keep the resulting alternatives.

Finish by matching the result to 'now facing', 'from the starting point', 'relative to the other person' or 'shortest distance'. These phrases select different outputs from the same diagram.