Mayank’s house faces South. He leaves from the back gate of his house and walks 18 m. Then he turns right and walks 28 m. Then he turns right and walks 35 m. Then he turns left and walks 12 m. He then turns left and walks 17 m. In which direction and how many meters away is he from the original position?

Question

Mayank&rsquo;s house faces South. He leaves from the back gate of his house and walks 18 m. Then he turns right and walks 28 m. Then he turns right and walks 35 m. Then he turns left and walks 12 m. He then turns left and walks 17 m. In which direction and how many meters away is he from the original position?

Prepp Team · Accepted Answer

Understanding Direction and Distance Problems

This question asks us to find the final position of a person, Mayank, relative to his starting point after a series of movements in different directions. To solve this type of problem, we can visualize the movements on a coordinate plane or simply track the net displacement in the East-West and North-South directions.

First, let's establish the starting point and direction. Mayank's house faces South, which means the back gate faces the opposite direction, North. He leaves from the back gate, so his initial movement is towards the North.

Let's break down Mayank's journey step by step:

He leaves from the back gate (facing North) and walks 18 m.
Then he turns right. When facing North, a right turn is towards the East. He walks 28 m East.
Then he turns right again. When facing East, a right turn is towards the South. He walks 35 m South.
Then he turns left. When facing South, a left turn is towards the East. He walks 12 m East.
He then turns left again. When facing East, a left turn is towards the North. He walks 17 m North.

Tracking Mayank's Movements

We can track the movements in terms of North (+ve Y-axis), South (-ve Y-axis), East (+ve X-axis), and West (-ve X-axis). Let's assume the starting point is the origin (0,0).

Step	Direction	Distance (m)	X Change	Y Change	Current Position (X, Y)
Start	-	-	0	0	(0, 0)
1	North	18	0	+18	(0, 18)
2	Right (East)	28	+28	0	(0+28, 18+0) = (28, 18)
3	Right (South)	35	0	-35	(28+0, 18-35) = (28, -17)
4	Left (East)	12	+12	0	(28+12, -17+0) = (40, -17)
5	Left (North)	17	0	+17	(40+0, -17+17) = (40, 0)

Alternatively, we can calculate the net displacement in the East-West and North-South directions directly:

North-South Movement:
- Starts North 18 m (+18)
- Turns South 35 m (-35)
- Turns North 17 m (+17)
- Net North-South displacement: $+18 - 35 + 17 = 0$ m.
East-West Movement:
- Turns East 28 m (+28)
- Turns East 12 m (+12)
- Net East-West displacement: $+28 + 12 = +40$ m.

Calculating Final Position and Distance

The net displacement is 0 m in the North-South direction and +40 m in the East-West direction. A positive value for East-West means he is 40 meters towards the East from the starting point. A net displacement of 0 in North-South means he is neither North nor South of the starting point.

His final position is effectively 40 meters East and 0 meters North/South of his original position.

The distance from the original position is the magnitude of the displacement vector, which in this case is simply the Eastward displacement since the North-South displacement is zero.

Distance $= \sqrt{(\text{Net East-West})^2 + (\text{Net North-South})^2}$

Distance $= \sqrt{(40)^2 + (0)^2}$

Distance $= \sqrt{1600 + 0}$

Distance $= \sqrt{1600}$

Distance $= 40$ meters.

Since the entire displacement is towards the East (+ve X-axis), his final direction from the original position is East.

Conclusion on Mayank's Final Position

After all the movements, Mayank is 40 meters away from his original position, and he is in the East direction relative to that position.

Revision Table: Key Concepts for Direction & Distance

Concept	Description
Cardinal Directions	North, South, East, West. Often mapped to Y+, Y-, X+, X-.
Turning Right/Left	Relative to the current direction of movement. If facing North, Right is East, Left is West. If facing East, Right is South, Left is North, etc.
Displacement	The shortest distance between the initial and final points, including direction. It's a vector quantity.
Distance Traveled	The total length of the path covered. This is a scalar quantity and different from displacement.
Original Position	The starting point, often considered the origin (0,0).
Final Position	The end point after all movements.

Additional Information on Direction Problems

Direction and distance questions often involve tracking movements on a 2D plane. It's helpful to:

Draw a diagram: Sketching the path can make it easier to visualize the movements and final position.
Use coordinates: Assigning coordinates (X, Y) to the starting point (e.g., (0,0)) and calculating the change in coordinates for each step helps find the final position systematically. North/South affects the Y-coordinate, and East/West affects the X-coordinate.
Net Displacement Calculation: Sum up all movements in the North-South direction and all movements in the East-West direction separately. The final position is determined by these net values.

Remember that "distance from the original position" usually refers to the straight-line distance (magnitude of displacement) between the start and end points.