Question

In a morning, Vivek started running from his home towards the Sun. He runs for 300 m, then again 300 m towards his right. Further runs 100 m towards his left to reach a Gym. His office is 800 m East of his home. What is the shortest distance and in which direction is his office from the Gym ?

• A. $700 \text{ m, North}$
• B. $500 \text{ m, North-East}$
• C. $300 \text{ m, North-West}$
• D. $700 \text{ m, South}$

Hint:

Remember the standard Pythagorean triplet $(3, 4, 5)$.
Here, the horizontal separation is $400 \text{ m}$ and the vertical separation is $300 \text{ m}$.
The hypotenuse must be $500 \text{ m}$, saving you the time of calculating square roots.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
This is a direction sense test problem.
We need to track the movement of a person step-by-step using a Cartesian coordinate system, locate his final destination (the Gym) and his office relative to his starting point (home), and calculate the shortest distance and direction between these two locations.

Step 2: Key Formula or Approach:
Let the starting point (home) be the origin $(0, 0)$.
East is represented along the positive x-axis, West along the negative x-axis, North along the positive y-axis, and South along the negative y-axis.
The shortest distance $d$ between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

Step 3: Detailed Explanation:

• Track Vivek's Movements:
- "In a morning, Vivek started running towards the Sun":
Since it is morning, the Sun is in the East. He runs East for $300 \text{ m}$.
Position 1 $= (300, 0)$.
- "then again 300 m towards his right":
Facing East, a right turn points South. He runs South for $300 \text{ m}$.
Position 2 $= (300, -300)$.
- "Further runs 100 m towards his left to reach a Gym":
Facing South, a left turn points East. He runs East for $100 \text{ m}$.
Position of Gym $(G) = (300 + 100, -300) = (400, -300)$.

• Determine the Position of the Office:
- "His office is 800 m East of his home":
Position of Office $(O) = (800, 0)$.

• Calculate the Distance and Direction of Office from Gym:
We need to find the vector from Gym $G(400, -300)$ to Office $O(800, 0)$:
\[ \Delta x = 800 - 400 = 400 \text{ m} \quad (\text{towards East}) \] \[ \Delta y = 0 - (-300) = 300 \text{ m} \quad (\text{towards North}) \] The shortest distance is:
\[ d = \sqrt{400^2 + 300^2} = \sqrt{160000 + 90000} = \sqrt{250000} = 500 \text{ m} \] Since $\Delta x$ is positive (East) and $\Delta y$ is positive (North), the direction of his office from the Gym is North-East.

Step 4: Final Answer:
The shortest distance is $500 \text{ m}$ and the direction is North-East.
Therefore, the correct option is (B).