Direction & Distance

Two people, one grid

Relative-position questions put two walkers (or a walker and a fixed object) on the same grid and ask how far apart they are, or in which direction one sits from the other. The method: give each person their own copy of the net-move bookkeeping, then compare.

Positions first. Fix the common start (or the named reference point) at the origin. Walk each person separately, netting their moves: person P ends at $(E_P, N_P)$, person Q at $(E_Q, N_Q)$.

Distance between them. The gap is just the distance between two grid points:

$d = \sqrt{(E_Q - E_P)^2 + (N_Q - N_P)^2}$

Most exam numbers land on clean triples (3-4-5, 5-12-13) or on a single axis after cancellation. If both walkers move along one shared leg ("both then walk 12 km East"), that shared leg contributes nothing to the gap — it shifts both equally.

Direction of one from the other. Stand at the person the question names second: the arrow to the other person has components $\Delta E = E_Q - E_P$, $\Delta N = N_Q - N_P$. Read the compass word of $(\Delta E, \Delta N)$ exactly as in final-direction questions. Sanity-check with a rough sketch — the answer word must match what the picture shows.

Fixed reference points. Many questions measure everything from an object (a pole, a well, a house) that itself never moves. That is easier, not harder: the reference is the origin, and each object's position is its own net walk from it.

Meeting points. If two people start together and walk for the same time, they meet where their positions coincide; questions often ask the distance of the meeting point from the start — walk each to the meeting coordinates and apply the distance formula on either one.

Two-person direction chains. "A is 5 km East of B; B is 12 km North of C" — chain them from the asked pair: place C at the origin, put B at (0, 12), put A at (5, 12). Then any question about A and C is one distance formula: 13 km. Chained questions are just single-walker questions with an extra setup step.

Word order matters. "Direction of A from B" and "direction of B from A" are opposite answers. Same for "A is East of B" (A sits on B's East) versus "B is East of A". Underline the reference person in every sentence before placing anyone — in chains, the person named after 'of' is the reference for that sentence.

When the answer is a word plus a number. Many questions ask "how far and in which direction" together. Compute the number first (gap formula), then the word (compass of the difference), and match the pair against the options as a unit — options often pair the correct distance with a wrong direction to catch half-solvers.

Quick revision

• Walk each person separately from the same origin; end with $(E, N)$ per person.
• Gap: $\sqrt{(\Delta E)^2 + (\Delta N)^2}$; shared legs cancel.
• "Direction of X from Y": compute $(E_X - E_Y, N_X - N_Y)$, read the compass word.
• Chains ("A of B, B of C"): place the last person at the origin and stack the offsets.
• Verify every answer against a rough sketch — the picture catches sign slips.