Vilas went from his home to his college. He started his journey facing East. First, he went 2 km straight; then he turned to his right and went 3 km; finally, he turned left and walked 4 km to reach his college.
North - west
Solving the Directional Reasoning Problem
The problem asks for the direction of Vilas's home relative to his college after a series of movements. To solve this, we need to track Vilas's movements step-by-step and determine the final position of the college relative to his starting point (home).
Step-by-Step Movement Analysis
Let's assume Vilas starts at his home at the origin (0,0) on a coordinate plane, where East is the positive x-axis, West is the negative x-axis, North is the positive y-axis, and South is the negative y-axis.
- Starting Point: Home (0,0). Vilas faces East.
- Movement 1: He goes 2 km straight facing East.
- Direction: East (+x)
- Distance: 2 km
- New position relative to home: (0 + 2, 0) = (2,0)
- Movement 2: He turns right. Since he was facing East, a right turn makes him face South. He then goes 3 km.
- Direction: South (-y)
- Distance: 3 km
- New position relative to home: (2, 0 - 3) = (2,-3)
- Movement 3: He turns left. Since he was facing South, a left turn makes him face East. He then walks 4 km to reach his college.
- Direction: East (+x)
- Distance: 4 km
- New position relative to home: (2 + 4, -3) = (6,-3)
So, the location of the college relative to his home is at coordinate (6, -3). This means the college is 6 km East and 3 km South of his home.
Finding Home's Direction Relative to College
The question asks for the direction of his home relative to his college. We found that the college is at (6, -3) if the home is at (0,0).
To find the position of the home relative to the college, we need to consider the displacement vector from the college back to the home. If the college is at position $\vec{C} = (6, -3)$ relative to the home $\vec{H} = (0,0)$, then the vector from college to home is $\vec{H} - \vec{C}$.
$\vec{H} - \vec{C} = (0,0) - (6,-3) = (0 - 6, 0 - (-3)) = (-6, 3)$
The position $(-6, 3)$ relative to the college means:
- -6 in the x-direction corresponds to 6 km West.
- +3 in the y-direction corresponds to 3 km North.
Therefore, the home is located 6 km West and 3 km North of the college. This relative position corresponds to the North-West direction.
Let's summarize the final position of college relative to home:
| Direction | Distance (km) | Total Displacement (km) |
|---|---|---|
| East | 2 + 4 | 6 (Positive x) |
| South | 3 | -3 (Negative y) |
College is at (6, -3) relative to Home (0,0).
Home is at (-6, 3) relative to College (0,0).
A position of (-6, 3) means West (negative x) and North (positive y), which is the North-West direction.
Final Answer Confirmation
Based on the movement analysis and calculating the relative position, Vilas's home is located in the North-West direction in relation to his college.
Revision Table: Key Concepts
| Concept | Explanation | Application in Problem |
|---|---|---|
| Directional Movement | Moving straight in a specific direction (N, S, E, W). | Vilas moved East, then South, then East. |
| Turns (Right/Left) | Changing facing direction by 90 degrees. Right from East is South; Left from South is East. | Used to determine direction after turns. |
| Relative Position | The location of one point with respect to another point. | Finding college location relative to home, then home location relative to college. |
| Coordinate System | Using (x,y) coordinates to represent position, with origin as reference. | Used to track displacement numerically (East > +x, West > -x, North > +y, South > -y). |
Additional Information: Understanding Relative Directions
When a question asks for the direction of point A relative to point B, it means if you are standing at point B, which direction do you need to face and move to reach point A?
- If point A is North of point B, its direction is North.
- If point A is West of point B, its direction is West.
- If point A is North-West of point B, its direction is North-West.
In this problem, we found that the home is at (-6, 3) relative to the college. This vector $(-6, 3)$ points 6 units in the negative x (West) direction and 3 units in the positive y (North) direction. Combining West and North gives the North-West direction.
Important Questions from Alphabet Series
Select the letter-cluster from among the given options that can replace the question mark (?) in the following series.
KMTC, EVOM, OQXG, IZSQ, ?
Select the set of letters that when sequentially placed in the blanks of the given letter series will complete the series.
k_lmml_mk_mmk_lkkl_mSelect the letter will replace the question mark (?) in the following series.
C, B, B, C, Z, E, W, H, S, ?, NWhich letter will replace the question mark (?) in the following letter series?
E, J, N, Q, S, ?
Select the combination of letters that when sequentially placed in the blanks of the given series will complete the series.
C _ B N _ _ V_ _ H C _ B _ H
