Mohit starts from point Y and drives 6 km towards south. He then takes a left turn, drives 23 km, turns right and drives 27 km. He then takes a right turn and drives 11 km. He takes a right turn, drives 13 km. He then turns right, drives 19 km, turns left and drives 20 km to stop at point Z. How far (shortest distance) and towards which direction should he drive in order to reach point Y again? (All turns are 90 degrees turns only unless specified.) 

Place Y at the origin (0,0)(0,0) ( 0 , 0 ) . Take north as +y+y + y , east as +x+x + x .

Follow Mohit’s moves step by step:

1. 6 km south → (0,−6)(0,-6) ( 0 , − 6 )

2. Left (facing south → left = east) 23 km → (23,−6)(23,-6) ( 23 , − 6 )

3. Right (facing east → right = south) 27 km → (23,−33)(23,-33) ( 23 , − 33 )

4. Right (facing south → right = west) 11 km → (12,−33)(12,-33) ( 12 , − 33 )

5. Right (facing west → right = north) 13 km → (12,−20)(12,-20) ( 12 , − 20 )

6. Right (facing north → right = east) 19 km → (31,−20)(31,-20) ( 31 , − 20 )

7. Left (facing east → left = north) 20 km → (31,0)(31,0) ( 31 , 0 ) = point Z

So Z is at (31,0)(31,0) ( 31 , 0 ) and Y is at (0,0)(0,0) ( 0 , 0 ) . The displacement from Z to Y is (−31,0)(-31,0) ( − 31 , 0 ) , i.e. 31 km due west.

Answer: He must drive 31 km towards the west to return to point Y.