In a river , a boat with a vertical pole on it is moving and velocity of river, boat and man climbing on the pole are given below
`barV_(1)=3 m//s hati,barV_j= 4m//s hatj,V_(mb)= 2m//s k`
Symbols r,b and m are used for river boat and man.Velocity of the man as observed by ground observer will be

Options

A. `3hati+4hatj+2hatk`
B. `4hatj+2hatk`
C. `2hatk`
D. `3hati+2hatk`

Explanation

To find the velocity of the man as observed by a ground observer, we need to consider the velocities of the river, the boat, and the man climbing the pole on the boat. We will use vector addition to determine the resultant velocity of the man.

### Step-by-Step Solution:

1. **Identify the Given Velocities:**
- Velocity of the river \( \vec{V}_r = 3 \, \text{m/s} \, \hat{i} \) (along the x-direction)
- Velocity of the boat \( \vec{V}_b = 4 \, \text{m/s} \, \hat{j} \) (along the y-direction)
- Velocity of the man with respect to the boat \( \vec{V}_{mb} = 2 \, \text{m/s} \, \hat{k} \) (along the z-direction)

2. **Determine the Velocity of the Man with Respect to the Ground:**
- The velocity of the man as observed from the ground \( \vec{V}_m \) can be calculated using the formula:
\[ \vec{V}_m = \vec{V}_{mb} + \vec{V}_b \]
- Here, \( \vec{V}_{mb} \) is the velocity of the man with respect to the boat, and \( \vec{V}_b \) is the velocity of the boat with respect to the ground.

3. **Calculate the Components of the Velocity:**
- The velocity of the man with respect to the ground can be expressed as:
\[ \vec{V}_m = \vec{V}_{mb} + \vec{V}_b + \vec{V}_r \]
- Since the boat is moving in the y-direction and the river is flowing in the x-direction, we can combine these velocities:
\[ \vec{V}_m = 4 \, \hat{j} + 2 \, \hat{k} + 3 \, \hat{i} \]

4. **Combine the Velocities:**
- Thus, the total velocity of the man as observed from the ground is:
\[ \vec{V}_m = 3 \, \hat{i} + 4 \, \hat{j} + 2 \, \hat{k} \]

5. **Final Result:**
- The velocity of the man as observed by the ground observer is:
\[ \vec{V}_m = 3 \, \hat{i} + 4 \, \hat{j} + 2 \, \hat{k} \, \text{m/s} \]