Two particles A and B move along the straight lines `x+2y+3=0` and `2x+y-3=0` respectively. Their positive vector, at the time of meeting will be

Options

A. `3hat(i)+3hat(j)`
B. `3hat(i)-3hat(j)`
C. `(hat(i))/3-(hat(j))/3`
D. Particles never meet

Explanation

To solve the problem, we need to find the point where the two particles A and B meet, given their respective paths defined by the equations of the lines they move along.

### Step-by-Step Solution:

1. **Identify the equations of the lines**:
- For particle A: The equation is \( x + 2y + 3 = 0 \).
- For particle B: The equation is \( 2x + y - 3 = 0 \).

2. **Rearrange the equations**:
- For particle A, rearranging gives:
\[ x = -3 - 2y \]
- For particle B, rearranging gives:
\[ y = 3 - 2x \]

3. **Substitute the expression for x from particle A into the equation for particle B**:
- Substitute \( x = -3 - 2y \) into \( y = 3 - 2x \):
\[ y = 3 - 2(-3 - 2y) \]
- Simplifying this:
\[ y = 3 + 6 + 4y \]
\[ y - 4y = 9 \]
\[ -3y = 9 \]
\[ y = -3 \]

4. **Find the corresponding x-coordinate**:
- Substitute \( y = -3 \) back into the equation for particle A:
\[ x = -3 - 2(-3) = -3 + 6 = 3 \]

5. **Determine the position vector at the time of meeting**:
- The position vector \( \mathbf{r} \) at the meeting point is:
\[ \mathbf{r} = 3 \hat{i} - 3 \hat{j} \]

6. **Final Answer**:
- The positive vector at the time of meeting is:
\[ \mathbf{r} = 3 \hat{i} - 3 \hat{j} \]