Question

A wife and husband are standing in one place. Husband goes \(3\) km towards South and turns right and goes \(2\) km. Wife goes \(2\) km towards East. The wife will be in which direction from the husband?

• A. North-East
• B. South-East
• C. North-West
• D. South-West

Hint:

In direction questions, fix the starting point and track final coordinates of each person.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
This is a direction sense and distance problem. We need to plot the movements of both husband and wife from their home (starting point) on a coordinate plane, and then find the distance and direction of the wife's office relative to the husband's office.

Step 2: Key Formula or Approach:
Let us set the starting point (home) at the origin $(0, 0)$.
• North represents positive Y-axis, South represents negative Y-axis.

• East represents positive X-axis, West represents negative X-axis.

• Distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

Step 3: Detailed Explanation:
Let us trace the paths:
• Husband's Path:
Starts at $(0, 0)$.
Drives 3 km South: reaches coordinates $(0, -3)$.
Turns to his right: Facing South, turning right means turning towards the West (negative X-direction).
Drives 2 km West: reaches final coordinates $(-2, -3)$.
So, Husband's Office ($H$) is at $(-2, -3)$.

• Wife's Path:
Starts at $(0, 0)$.
Drives 2 km East (positive X-direction): reaches coordinates $(2, 0)$.
So, Wife's Office ($W$) is at $(2, 0)$.

• Find distance and direction of $W$ with respect to $H$:
We need the vector from $H(-2, -3)$ to $W(2, 0)$: \[ \Delta x = x_W - x_H = 2 - (-2) = 4\text{ km} \] \[ \Delta y = y_W - y_H = 0 - (-3) = 3\text{ km} \] Using Pythagoras theorem: \[ d = \sqrt{(\Delta x)^2 + (\Delta y)^2} = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5\text{ km} \] Since $\Delta x = +4$ (East) and $\Delta y = +3$ (North), the direction from the husband's office to the wife's office is North-East.

Step 4: Final Answer:
The wife's office is 5 km, North-east with respect to the husband's office. This corresponds to option (A).