Abhijeet leaves his house and moves 30 m in the North-West direction. Then, he goes 30 m in the South-West direction. Next, he moves 30 m in the South-East direction. Then, he moves 30 m in the North-East direction. How far away is he from his house?
0 m
Understanding the Problem: Distance from Starting Point
The question asks for the final distance of Abhijeet from his house after a series of movements. He starts at his house and makes four distinct moves, each covering a distance of 30 m in a specific diagonal direction.
To solve this, we need to understand how movements in different directions combine. We can think of each diagonal movement as having components in the North-South and East-West directions.
Analyzing Each Movement and its Components
Let's assume the house is at the origin (0,0). We can break down each movement into its East (positive x) and North (positive y) components. Movements West are negative x, and movements South are negative y.
- North-West (NW): This direction is exactly between North and West. A 30 m movement in this direction means moving 30 m multiplied by $\cos(45^\circ)$ towards West and 30 m multiplied by $\sin(45^\circ)$ towards North. Since $\cos(45^\circ) = \sin(45^\circ) = \frac{1}{\sqrt{2}}$, the components are $-30 \times \frac{1}{\sqrt{2}}$ in the East direction and $+30 \times \frac{1}{\sqrt{2}}$ in the North direction.
- South-West (SW): This direction is exactly between South and West. A 30 m movement means moving $30 \times \frac{1}{\sqrt{2}}$ towards West and $30 \times \frac{1}{\sqrt{2}}$ towards South. The components are $-30 \times \frac{1}{\sqrt{2}}$ in the East direction and $-30 \times \frac{1}{\sqrt{2}}$ in the North direction.
- South-East (SE): This direction is exactly between South and East. A 30 m movement means moving $30 \times \frac{1}{\sqrt{2}}$ towards East and $30 \times \frac{1}{\sqrt{2}}$ towards South. The components are $+30 \times \frac{1}{\sqrt{2}}$ in the East direction and $-30 \times \frac{1}{\sqrt{2}}$ in the North direction.
- North-East (NE): This direction is exactly between North and East. A 30 m movement means moving $30 \times \frac{1}{\sqrt{2}}$ towards East and $30 \times \frac{1}{\sqrt{2}}$ towards North. The components are $+30 \times \frac{1}{\sqrt{2}}$ in the East direction and $+30 \times \frac{1}{\sqrt{2}}$ in the North direction.
Step-by-Step Displacement Calculation
Let's add up the components of each movement to find the total displacement from the starting point (house). We can represent the components as $(\Delta x, \Delta y)$. Let $d = \frac{30}{\sqrt{2}}$.
| Movement | Direction | Distance (m) | East-West Component ($\Delta x$) | North-South Component ($\Delta y$) |
|---|---|---|---|---|
| 1st | North-West | 30 | $-\frac{30}{\sqrt{2}} = -d$ | $+\frac{30}{\sqrt{2}} = +d$ |
| 2nd | South-West | 30 | $-\frac{30}{\sqrt{2}} = -d$ | $-\frac{30}{\sqrt{2}} = -d$ |
| 3rd | South-East | 30 | $+\frac{30}{\sqrt{2}} = +d$ | $-\frac{30}{\sqrt{2}} = -d$ |
| 4th | North-East | 30 | $+\frac{30}{\sqrt{2}} = +d$ | $+\frac{30}{\sqrt{2}} = +d$ |
Now, let's find the total displacement by summing the components:
- Total East-West Displacement ($\Delta X_{total}$): $(-d) + (-d) + (+d) + (+d) = -2d + 2d = 0$
- Total North-South Displacement ($\Delta Y_{total}$): $(+d) + (-d) + (-d) + (+d) = 0d + 0d = 0$
The final position relative to the starting point (house) is $(0,0)$.
Calculating the Final Distance from the House
The final position is at the coordinates $(0,0)$. The starting point (house) was also at $(0,0)$. The distance between the final position and the starting position is the magnitude of the total displacement vector $(\Delta X_{total}, \Delta Y_{total})$.
Distance $= \sqrt{(\Delta X_{total})^2 + (\Delta Y_{total})^2}$
Distance $= \sqrt{(0)^2 + (0)^2} = \sqrt{0 + 0} = \sqrt{0} = 0$ m.
Therefore, Abhijeet is 0 m away from his house. He has returned to his starting point.
Revision Table: Direction and Displacement Concepts
| Concept | Description |
|---|---|
| Distance | The total length covered during a movement, regardless of direction. It is a scalar quantity. |
| Displacement | The straight-line distance and direction from the starting point to the ending point. It is a vector quantity. |
| Components of Displacement | Breaking down a movement into its horizontal (East-West) and vertical (North-South) parts. Useful for calculating total displacement from multiple movements. |
Additional Information: Path vs. Displacement
It is important to distinguish between the total distance traveled and the final displacement. In this problem, Abhijeet traveled a total distance of $30 m + 30 m + 30 m + 30 m = 120 m$. However, because he ended up back at his starting point, his total displacement from the house is 0 m. The question specifically asks how far away he is from his house, which refers to the magnitude of the displacement from the starting point.
The sequence of movements forms a closed loop, bringing Abhijeet back to where he began. Visualizing this path on a coordinate system or as a series of vectors starting from the origin confirms that the tail of the first vector is at (0,0) and the head of the last vector also ends at (0,0).
Important Questions from Direction and Distance
- Vijendra walks a certain distance, say X metres, from his home towards the west. Then he turns left and walks 23 metres. After that, he turns left and walks 36 metres. Then he turns left again to walk 23 metres. He finally turns left and walks 18 metres to reach his home. Find the value of X.
Gaurav exits from the backdoor of his north-facing house and walks 25 m straight, then he takes a left turn and walks 36 m, then he turns left and walks 47 m. He turns left again and walks 36 m. How far and in which direction is he from his house now?
Reeta is standing facing south-east. First, she turns 135° clockwise. After that, she turns 90 ° anticlockwise. Then she turns 45 ° clockwise, followed by a 135 ° anticlockwise turn. In which direction is she facing now?
Sahasra started running from her house towards the north. After 40 meters she turned left and ran for 75 meters. She then turned right and ran for 30 meters, and again turned right to run 75 meters. In which direction was she running finally?
At the time of sunset, Lopa and Kritika are sitting facing each other. If the shadow of Lopa falls to the right of Kritika, in which direction is Kritika facing?
