A boat is moving towards east velocity 4 m/s w.r.t still water and river is flowing towards north with velocity 2 m/s and the wind blowing towards north with velocity 6 m/s. the direction of the flag blown over by the wind hpisted on the boat is

Options

A. a. North - west
B. b. south- west
C. c. `tan^(-1) (1/2)` with east
D. d. North

Answer

To solve the problem, we need to analyze the velocities of the boat, river, and wind, and then determine the direction of the flag blown by the wind on the boat.

### Step-by-Step Solution:

1. **Define the Coordinate System:**
- Let the east direction be the positive x-axis and the north direction be the positive y-axis.

2. **Identify Given Velocities:**
- Velocity of the boat with respect to still water, \( V_{B} = 4 \, \text{m/s} \) towards east:
\[ V_{B} = 4 \hat{i} \]
- Velocity of the river flowing towards north, \( V_{R} = 2 \, \text{m/s} \):
\[ V_{R} = 2 \hat{j} \]
- Velocity of the wind blowing towards north, \( V_{W} = 6 \, \text{m/s} \):
\[ V_{W} = 6 \hat{j} \]

3. **Calculate the Velocity of the Boat with Respect to the River:**
- The velocity of the boat with respect to the river is given as \( V_{B/R} = 4 \hat{i} \).
- Thus, the velocity of the boat with respect to the ground can be expressed as:
\[ V_{B/G} = V_{B/R} + V_{R} = 4 \hat{i} + 2 \hat{j} \]

4. **Calculate the Velocity of the Wind with Respect to the Boat:**
- To find the velocity of the wind with respect to the boat, we subtract the velocity of the boat from the velocity of the wind:
\[ V_{W/B} = V_{W} - V_{B/G} \]
- Substitute the values:
\[ V_{W/B} = 6 \hat{j} - (4 \hat{i} + 2 \hat{j}) = -4 \hat{i} + 4 \hat{j} \]

5. **Determine the Direction of the Resultant Velocity:**
- The resultant velocity vector \( V_{W/B} = -4 \hat{i} + 4 \hat{j} \) indicates that the wind is blowing towards the northwest direction (since it has a negative x-component and a positive y-component).
- To find the angle of this vector, we can use the tangent function:
\[ \tan(\theta) = \frac{4}{4} = 1 \implies \theta = 45^\circ \]
- Thus, the direction of the flag blown by the wind on the boat is at an angle of \( 45^\circ \) from the negative x-axis towards the positive y-axis, which is northwest.

### Conclusion:
The direction of the flag blown over by the wind, which is posted on the boat, is towards the northwest.