Following three forces keep a body in equilibrium. `vecF_1=hati+3hatj+2hatk , vecF_2=3hati-4hatk` and `vecF_3=ahati-3hatj+2hatk`, then the value of a is

Options

A. 1
B. `-1`
C. 2
D. `-4`

Explanation

To solve the problem, we need to find the value of \( a \) such that the three forces \( \vec{F_1} \), \( \vec{F_2} \), and \( \vec{F_3} \) keep the body in equilibrium. The condition for equilibrium is that the sum of the forces must equal zero.

### Step-by-Step Solution:

1. **Write down the forces**:
\[ \vec{F_1} = \hat{i} + 3\hat{j} + 2\hat{k} \]
\[ \vec{F_2} = 3\hat{i} - 4\hat{k} \]
\[ \vec{F_3} = a\hat{i} - 3\hat{j} + 2\hat{k} \]

2. **Set up the equilibrium condition**:
For the body to be in equilibrium, the sum of the forces must equal zero:
\[ \vec{F_1} + \vec{F_2} + \vec{F_3} = 0 \]

3. **Combine the forces**:
Substitute the expressions for \( \vec{F_1} \), \( \vec{F_2} \), and \( \vec{F_3} \):
\[ (\hat{i} + 3\hat{j} + 2\hat{k}) + (3\hat{i} - 4\hat{k}) + (a\hat{i} - 3\hat{j} + 2\hat{k}) = 0 \]

4. **Group the components**:
Combine the \( \hat{i} \), \( \hat{j} \), and \( \hat{k} \) components:
- For \( \hat{i} \):
\[ 1 + 3 + a = 4 + a \]
- For \( \hat{j} \):
\[ 3 - 3 = 0 \]
- For \( \hat{k} \):
\[ 2 - 4 + 2 = 0 \]

5. **Set the components equal to zero**:
Since the sum must equal zero, we can set the \( \hat{i} \) component equal to zero:
\[ 4 + a = 0 \]

6. **Solve for \( a \)**:
Rearranging the equation gives:
\[ a = -4 \]

### Final Answer:
The value of \( a \) is \( -4 \).