A man travels 10km towards west, then he takes a turn towards north-east and travels 20km in that direction. He again takes a turn towards south-east and covers 5km. Taking the starting point as the origin, in which direction is he now?

Let's map out the movement steps coordinateswise starting from the origin (0,0):
Step 1: Moves 10km West.
New Position A=(−10,0).

Step 2: Turns North-East and travels 20km.
North-East direction forms an angle of 45
∘
with the positive x-axis (East/North).
The change in coordinates is:

Δx=20cos(45
∘
)=20×
2

​

1
​
≈14.14km
Δy=20sin(45
∘
)=20×
2

​

1
​
≈14.14km
New Position B=(−10+14.14,0+14.14)=(4.14,14.14).

Step 3: Turns South-East and covers 5km.
South-East direction means moving at −45
∘
(or 315
∘
) relative to the East direction line.

Δx=5cos(−45
∘
)=5×
2

​

1
​
≈3.54km
Δy=5sin(−45
∘
)=−5×
2

​

1
​
≈−3.54km
Final Position C=(4.14+3.54,14.14−3.54)=(7.68,10.60).

Since both the final x-coordinate (7.68) and y-coordinate (10.60) are positive, the person is located in the first quadrant relative to the origin (0,0), which corresponds to the North-east direction.