Step 1: Understanding the Question:
This problem tracks the net displacement of a person's facing direction after a series of angular rotations.
Standard directions follow a \(360^{\circ}\) circle: North (\(0^{\circ}\)), East (\(90^{\circ}\)), South (\(180^{\circ}\)), and West (\(270^{\circ}\)).
Step 2: Key Formula or Approach:
The most efficient way is to calculate the net turn:
\[ \text{Net Turn} = (\text{Sum of Clockwise Turns}) - (\text{Sum of Counterclockwise Turns}) \]
A positive result means a net clockwise turn, and a negative result means a net counterclockwise turn.
Step 3: Detailed Explanation:
Initial Direction: The man is facing West. On the compass, West is \(270^{\circ}\) from North.
List of Turns:
1. Turn 1: \(+45^{\circ}\) (Clockwise)
2. Turn 2: \(+180^{\circ}\) (Clockwise)
3. Turn 3: \(-270^{\circ}\) (Counterclockwise)
Calculating Net Displacement:
Net Turn = \(45 + 180 - 270\)
Net Turn = \(225 - 270\)
Net Turn = \(-45^{\circ}\)
Interpreting the Result: A net turn of \(-45^{\circ}\) means the person has turned \(45^{\circ}\) counterclockwise from his original starting position.
Applying to Initial Direction: Starting from West and moving \(45^{\circ}\) counterclockwise (towards the South) leads to the direction South-West.
Verification via Angles: Initial position = \(270^{\circ}\). Final position = \(270^{\circ} - 45^{\circ} = 225^{\circ}\). On a standard compass, \(225^{\circ}\) is exactly South-West.
Step 4: Final Answer:
The man is now facing the South-West direction. Option (B) is the correct answer.