For four particle A,B,C,D, the velocities of one with respect to other are given as `vecV_(DC)` is `20(m)/(s)` towards north, `vecV_(BC)` is `20(m)/(s)` towards east and `vecV_(BA)` is `20(m)/(s)` towards south. Then `vecV_(DA)` is

Options

A. `20m//s` towards north
B. `20m//s` towards south
C. `20m//s` towards east
D. `20m//s` towards west

Answer

To solve the problem of finding the velocity of particle D with respect to particle A (denoted as \( \vec{V}_{DA} \)), we can use the given velocities of the particles with respect to each other.

### Given:
1. \( \vec{V}_{DC} = 20 \, \text{m/s} \) towards North
2. \( \vec{V}_{BC} = 20 \, \text{m/s} \) towards East
3. \( \vec{V}_{BA} = 20 \, \text{m/s} \) towards South

### Step-by-step Solution:

**Step 1: Define the Directions**
- Let's assign a coordinate system:
- North = positive \( j \) direction
- East = positive \( i \) direction
- South = negative \( j \) direction
- West = negative \( i \) direction

**Step 2: Express the Given Velocities in Vector Form**
- \( \vec{V}_{DC} = 20 \, \hat{j} \) (North)
- \( \vec{V}_{BC} = 20 \, \hat{i} \) (East)
- \( \vec{V}_{BA} = -20 \, \hat{j} \) (South)

**Step 3: Use the Relative Velocity Formula**
- We know that:
\[ \vec{V}_{DA} = \vec{V}_{D} - \vec{V}_{A} \]
- We can express \( \vec{V}_{D} \) in terms of \( \vec{V}_{C} \) and \( \vec{V}_{A} \):
\[ \vec{V}_{DA} = \vec{V}_{D} - \vec{V}_{C} + \vec{V}_{C} - \vec{V}_{B} + \vec{V}_{B} - \vec{V}_{A} \]
- This simplifies to:
\[ \vec{V}_{DA} = \vec{V}_{DC} + \vec{V}_{CB} + \vec{V}_{BA} \]

**Step 4: Calculate \( \vec{V}_{CB} \)**
- Since \( \vec{V}_{BC} = 20 \, \hat{i} \), then:
\[ \vec{V}_{CB} = -\vec{V}_{BC} = -20 \, \hat{i} \]

**Step 5: Substitute the Values**
- Now we can substitute the values into the equation:
\[ \vec{V}_{DA} = \vec{V}_{DC} + \vec{V}_{CB} + \vec{V}_{BA} \]
\[ \vec{V}_{DA} = (20 \, \hat{j}) + (-20 \, \hat{i}) + (-20 \, \hat{j}) \]
\[ \vec{V}_{DA} = -20 \, \hat{i} + 0 \, \hat{j} \]

**Step 6: Final Result**
- Therefore, the velocity \( \vec{V}_{DA} = -20 \, \hat{i} \), which means:
\[ \vec{V}_{DA} = 20 \, \text{m/s} \text{ towards West} \]

### Conclusion:
The final answer is \( \vec{V}_{DA} = 20 \, \text{m/s} \) towards West.