Q1 A man walks 3 km north, then 4 km east. How far is he from his starting point?
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5 km.
The two legs are at right angles, so the direct distance is √(3² + 4²) = √25 = 5 km. He is north-east of where he began.
Direction Sense
What it is
These are coordinate geometry dressed up as a walk. Put north at the top of your sheet, draw each leg, and the final answer is almost always a right-angled triangle between the start and the end.
The one thing to be careful about is turns: "turns left" is relative to the direction the person is currently facing, not to the page.
What you need to know
- Draw north up, east right. Every leg is a line in one of four directions.
- Shortest distance between start and end is √(east-west² + north-south²).
- A left turn is anticlockwise from the current heading; a right turn is clockwise.
- At sunrise your shadow falls west; at sunset it falls east. This is the whole of the "shadow" variant.
- Opposite directions cancel. Add up all the north legs, subtract the south, and do the same east to west, before drawing anything.
Worked examples
Try each one before opening the answer. A solution you have read is not a solution you can produce under time pressure, which is the only kind that counts.
Q2 Walking 10 m south, then 6 m west, then 10 m north, then 6 m east: where does he end up?
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Back at the start.
Add the legs by axis. North-south: 10 south and 10 north cancel. East-west: 6 west and 6 east cancel. Both totals are zero, so the displacement is zero - he has walked a rectangle.
Q3 Facing north, a man turns right, walks 5 km, turns right again and walks 12 km. How far is he from the start, and in which direction?
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13 km, south-east.
Facing north and turning right makes him face east; he walks 5 km east. Turning right again makes him face south; he walks 12 km south. The distance is √(5² + 12²) = 13 km, and he is east and south of the start.
Where people lose marks
- Turning relative to the page instead of to the direction being faced.
- Adding the legs instead of using Pythagoras. The question asks for displacement, not distance walked.
- Mixing up which way a shadow falls. Morning sun is in the east, so shadows point west.