Direction & Distance

The one idea behind every distance question

Every direction question is a walk on a plain grid. North is up, East is right, and every sentence moves you by some amount. If you convert each sentence into a move on the grid, the answer stops being a puzzle and becomes arithmetic.

Set up the grid. Put the start at the origin and draw two axes: North up, East right. Give the walker a name and a pencil point. Each sentence of the question moves the point:

• "walks 5 km North" → 5 up.
• "turns left and walks 3 km" → depends on the way he was facing. If he was walking North, left is West → 3 to the left.
• "walks 5√2 km North-East" → 5 up and 5 right at the same time (a NE step of length k moves k/√2 in N and k/√2 in E; exams usually give it as k√2 so the split is exact).

Displacement vs distance. Two different numbers get swapped as options:

• Distance travelled = add every leg, ignoring direction.
• Displacement (shortest distance) = the straight line from start to finish.

Shortest distance in one step. Collect the net moves: net East = sum of rightward legs minus leftward legs; net North = sum of upward legs minus downward legs. Then

$d = \sqrt{(\text{net E})^2 + (\text{net N})^2}$

Cancellation comes first. Before any square root, cancel opposite legs: 5 km N then 5 km S is zero movement. Most exam questions hide 2-3 such cancellations. Cancel, then apply the formula on what remains — often you land on a clean triple: 3-4-5, 5-12-13, 8-15-17, 7-24-25.

Diagonal legs. A leg of k√2 along NE splits into k N and k E; along NW it is k N and k W; SE is k S + k E; SW is k S + k W. So "5√2 NE then 5√2 NW" = 10 N exactly, because the East and West parts cancel.

Do not rotate the map. Keep North up on paper always, even when the walker turns. The walker turns; your page never does. Mixing these two is the top cause of wrong answers.

Reading the question twice. Four phrasings hide in this subtopic, and each wants a different number:

• "How far is he from the start?" → displacement (Pythagoras on the net legs).
• "How much distance did he cover/walk?" → sum of all legs.
• "How much shorter would a straight road be?" → total minus displacement.
• "In which direction is he from the start?" → the compass word of the net legs.

Speed habits for the exam. Write the legs as a running tally (+3 E, +4 N, −12 N) instead of drawing every metre; the tally is the net displacement with no diagram at all. Check the final root against the triples — if the legs net to 9 and 12, the triple 9-12-15 saves the square root. And keep answers in the option's own form: if options say 2√5, do not convert to 4.47.

Quick revision

• Draw axes: N up, E right; start at the origin; one arrow per sentence.
• Cancel opposite legs first; then $d=\sqrt{E^2+N^2}$ on the leftovers.
• Distance travelled = sum of legs; shortest distance = displacement — different options.
• k√2 along a diagonal = k on each of its two component directions.
• Know triples 3-4-5, 5-12-13, 8-15-17, 7-24-25 (and multiples) on sight.