Direction and Distance

Most walking questions end with one demand: "How far, and in which direction, is he from the starting point?" The answer is the straight line from the start to the end. It depends only on where he finally stands, not on the order of the moves.

Why it is easy: every leg runs along North, South, East or West, so the start and the end are two corners of a right triangle. Its sides are the net North-South and the net East-West movement.

[[fig:0]]

The method:

• Track the facing after each turn (right = clockwise, left = anticlockwise).
• Add the North legs and subtract the South legs; do the same for East and West. Only opposites cancel.
• Distance = √(net NS² + net EW²). Look for a triple: 3-4-5, 5-12-13, 8-15-17, 7-24-25, 20-21-29 and their multiples.
• Direction: put the two nets together. Net North and net East is North-East; if one net is zero, the answer is a single main direction.

Ishaan walks 7 km West and 24 km North: 7² + 24² = 625, so he is 25 km away, to the North-West.

Distance = √(net NS² + net EW²); direction = where the two nets point.

Diagonal legs. If a turn is 45° or 135°, a leg may run North-East and so on. Break such a leg into its two axis parts: a diagonal leg of d√2 gives exactly d along each axis. Never assume a right angle at a 45° or 135° corner; the right triangle appears only at the end, from the nets.

[[fig:1]]

In the second picture Meena walks 5 km East, turns 135° clockwise (she now faces South-West) and walks 5√2 km. That is 5 West and 5 South, so she stands 5 km due South of the start. After 12 km East she is 13 km South-East of the start.

This is the most repeated question of the Railway papers. A long story: a person starts from point A, drives four or five legs with 90° turns, stops at P, and the last line asks: "How far (shortest distance) and in which direction must he go to reach A again?"

The distance is the straight distance from A to P. Only the direction changes: it is the opposite of the direction of P from A. If P is North-West of A, the way back is South-East.

Why it is easy: with 90° turns, legs 1, 3, 5 lie on one axis and legs 2, 4 on the other. So the whole story is two small sums, one along each axis.

[[fig:0]]

In the picture Karan walks 12 km East and then takes three left turns with 20, 27 and 12 km. He ends 15 km West and 8 km North of the start: 17 km (8-15-17) North-West, so he must return 17 km South-East. In all he has walked 12 + 20 + 27 + 12 = 71 km.

Three more forms use the same grid:

• Other pairs of points. Put the start at (0, 0), East as +x and North as +y, write the coordinates of every point, and the distance is √(dx² + dy²). The same works for two persons who start together.
• Total distance is the plain sum of the legs: no figure, no root.
• Look-alikes. If the last line asks for speed or time, it is not a compass question. Average speed = total distance ÷ total time; for equal out-and-back distances it is 2ab ÷ (a + b), not the simple average.

The way back is opposite to the direction of the end point; total distance is the plain sum of the legs.

Here the question is not about distance at all. It asks one thing: after all these turns, which way is the person facing? Forget the lengths and watch the turns.

A right turn is a quarter circle clockwise (90°); a left turn is a quarter circle anticlockwise. After four turns of the same kind you are back where you began, so every group of four cancels.

[[fig:0]]

Turns of 90°. Count the rights, count the lefts, and take net turns = rights − lefts. Remove groups of four:

A negative net is read as lefts. The order of the turns never matters, only how many of each kind there are. Always rotate from the given starting direction: from North-East one right turn gives South-East, not East.

Turns in degrees. When the turns are 45°, 135°, 180°, 225° …, give every direction a number of 45°-steps clockwise from North.

[[fig:1]]

N 0, NE 1, E 2, SE 3, S 4, SW 5, W 6, NW 7. A clockwise turn adds steps (90° = 2, 135° = 3, 180° = 4, 225° = 5, 270° = 6) and an anticlockwise turn subtracts them. Add everything and remove groups of 8.

Example: facing East (2), turn 135° clockwise (+3), then 90° anticlockwise (−2): 2 + 3 − 2 = 3, which is South-East.

Net turns = rights − lefts, reduced by 4; for degrees, count 45°-steps and reduce by 8.

Some questions run the film backwards. You are told where the person is facing at the end and asked where he was facing when he began. The turns are given, the start is hidden.

Think of what the turns did. If together they turned him by one right turn, then the start must be one right turn behind the final direction. To get back to the start, undo the net turn: apply it the opposite way to the final direction.

[[fig:0]]

The method:

• Find the net turns as in Type 3: rights − lefts, reduced by 4.
• Reverse it. Net 0: the start is the final direction itself. Net 1 right: one left from the final direction. Net 2: the opposite direction. Net 3 rights: one right from the final direction. (A net of lefts works the other way round.)
• Run forward from your answer to check. It takes ten seconds and removes the commonest slip.

In the picture a person ends facing East after a net of one right turn. His start was one left turn from East, which is North. Running forward, North plus one right turn is East. It fits.

Only right turns. The newest form says "she took 16 right turns, no left turn" or "he turned 5 times right". Then the net is that number: 16 is four full circles, so the start is the final direction itself; 5 is 4 + 1, so the start is one left turn from the final direction.

The examiner's decoy is the answer you get by applying the leftover turn forwards (East plus one right = South). It is always among the options, so the forward check matters.

Start = the final direction turned the opposite way by the net turns.

For turns in degrees do the same in 45°-steps: start = final steps − net steps.

Here several people, places or towns are placed around one another by statements, and you must say where one stands with respect to another. No walker is involved. The first job is to see the whole picture on a grid.

The method is a map in coordinates:

• Take one person as the origin (0, 0).
• Place every other person from the statements: "X is 12 m North of Q" puts X at (0, 12); "C is 5 m East of Q" puts C at (5, 0).
• For "A from C", subtract: A's coordinates minus C's coordinates. This gives how far East or West and North or South A lies from C, and Pythagoras gives the straight distance.

[[fig:0]]

In the picture, A from C is (0 − 5, 12 − 0) = 5 West and 12 North, so A is 13 m North-West of C. "A from C" starts at C and ends at A. For C from A the answer is the opposite, South-East.

Chains without distances are the usual Railway form: "K is east of L, M is south of K, N is west of M, O is north of N, L is north-west of O." Write only signs. Each statement moves one person along one axis, and a diagonal statement such as "L is north-west of O" is not decoration: it tells you the signs of both gaps. A rough sketch is enough, since only directions matter.

Cars in a jam. If cars are described as right, left, behind or in front of one another, take them all as facing North, as on a map: right = East, left = West, behind = South.

Coded directions. "P @ Q means P is 6 m North of Q": the key is printed in the question and changes every time. Translate each coded line into a position, then finish as above.

Fix one person at (0, 0); "X from Y" means the arrow starts at Y.

Here the names of the directions are shuffled by a statement: "If South-East is called North, North-East is called West and so on …". The directions themselves do not move; only their names change, and they change together, as if the whole compass had been rotated by the same amount. One given pair fixes the rotation for the whole compass.

[[fig:0]]

The method, using angles (N 0°, NE 45°, E 90°, SE 135°, S 180°, SW 225°, W 270°, NW 315°):

• Rotation = angle of the new name − angle of the real direction. "South-East is called North": 0° − 135° = −135°.
• New name of any direction = its angle + rotation. North-East: 45° − 135° = −90° = 270°, which is West.
• To find which real direction carries a given name, subtract the rotation.
• Check with a second pair if the question gives one. Here North-East becomes West, exactly as stated.

Faulty compass. "A needle which used to show South now points East" is the same idea. Where the faulty compass reads "South", the real direction is East, so the reading is the real direction plus 90°. To find the real direction from a reading, subtract that 90°.

New name = real direction + rotation; real direction = name − rotation.

Never rotate only the direction you are asked about; all of them move together, in the same sense. A 45° rotation is easily mistaken for 90° because the diagonals land on main directions, so work with angles and not with words.

A shadow always falls on the side opposite to the Sun. That one fact solves every shadow question. The Sun rises in the East and sets in the West, so:

• Just after sunrise the Sun is in the East and every shadow falls towards the West.
• Just before sunset the Sun is in the West and every shadow falls towards the East.

[[fig:0]]

Now the person. The question tells you on which side of him the shadow falls (his right, his left, in front, behind) and asks which way he faces. Work in two steps:

1. Write the compass direction of the shadow: morning West, evening East.
2. Put that direction on the stated side of the person. In the morning, if the shadow is on his right, his right hand points West, so he faces South (facing South, the right hand is West).

The table lets you check your reasoning:

Walking forms. "Going from his house to the office early in the morning, his shadow fell on his left. In which direction is the house from the office?" The shadow gives the direction he walks (morning, left: North). He walks away from the house, so the house lies behind him: it is South of the office.

Face to face. Two persons facing each other face opposite directions, and left and right belong to each person. Solve for the person whose side is named, then reverse for the other. A river or a wall works the same way: fix its compass line first, then say on which side of the person it appears.

[[fig:1]]

Shadow is opposite to the Sun: morning West, evening East. Then place it on the stated side of the person.

This works at sunrise and sunset only. At noon the Sun is overhead and the shadow is very short.

These four forms are not seen often, but each is settled in two lines once you know its one idea. Recognise them by their key word.

1. On his head. Turn a person upside down about the line he faces: his face still points the same way, but his left and right hands change sides. So for a head-stand, right hand = the left side of the face direction, and left hand = the right side. Facing North, his left hand points East.

[[fig:0]]

2. Train, smoke and wind. The smoke trails behind the train, so it starts opposite to the train's direction. The wind adds its push towards the direction it blows to ("wind from the South" blows towards the North). Join the two: West + North = North-West. If the two are 90° apart, the answer is midway between them.

[[fig:1]]

3. Clock laid flat. Fix one mark of the dial to a compass direction; the dial then runs clockwise like the compass. Every 3 hours is 90°, and every hour mark is 30°. The minute hand points at the mark (minutes ÷ 5); the hour hand moves 30° per hour.

[[fig:2]]

4. Roundabout and junction. Left and right are the traveller's own. Indian traffic keeps left, so it goes clockwise round a roundabout. Count the exits in that order from the arm of entry (the first exit is the first arm you meet), unless the question says otherwise. After the exit he heads away from the centre along that arm.

[[fig:3]]

Find the key word, apply its one idea, and check with a quick sketch.

These are exam conventions, not physics, so read the convention printed in the question first.