Direction Sense

Draw a small compass in the corner of the rough sheet before reading the question: north at the top, south at the bottom, east on the right, west on the left. Then trace the walk step by step, marking the facing direction after every turn.

1. Net displacement. Add the north–south moves against each other and the east–west moves against each other; what is left is one move along each axis. The straight-line distance is then the hypotenuse, so keep 3-4-5, 5-12-13 and 6-8-10 ready.
2. Facing direction after turns. A right turn is 90° clockwise and a left turn 90° anticlockwise. Mark the arrow after each instruction rather than at the end.
3. Shadows. In the early morning the sun is in the east, so shadows fall towards the west; in the evening the sun is in the west and shadows fall east. At noon the shadow is too short to fix a direction.
4. Left and right of a person. These are always taken from that person's facing direction, not from where you sit.

The single most common mistake is applying left and right from your own point of view after the walker has turned around. Once a person faces south, their left hand points east, not west. How to avoid it: redraw the facing arrow after every single turn, and read the next left or right off the arrow.

Also remember: the straight-line distance must always be smaller than the total distance walked and larger than either single leg — a check that catches an arithmetic slip in one second.

Draw the compass first: N up, S down, E right, W left · right turn = 90° clockwise, left = anticlockwise · net move on each axis, then distance = √(a² + b²); keep 3-4-5, 5-12-13, 6-8-10 ready · morning sun east → shadow west · left and right belong to the walker, not to you

A man walks 4 km towards the north, turns right and walks 3 km. How far is he from his starting point, and in which direction? → After the right turn he faces east, so the two legs are 4 km north and 3 km east; they lie on different axes and cannot cancel. Distance = √(4² + 3²) = √25 = 5 km, to the north-east. Sanity check: 5 is less than the 7 km he actually walked and more than either single leg of 4 and 3, exactly as a hypotenuse must be.