If three forces `2 hat i +3 hat j - hatk` and `3 hat I -18hat j -4hatk` and `a( hat i -3 hat j - hatk) ` hold a particle in equilibrium .Then a is

Options

A. 5
B. `-5`
C. 10
D. 15

Explanation

To solve the problem, we need to find the value of \( a \) such that the three forces acting on a particle result in equilibrium. This means that the sum of the forces must equal zero.

### Step-by-Step Solution:

1. **Identify the Forces**: We have three forces given:
- \( \mathbf{F_1} = 2\hat{i} + 3\hat{j} - \hat{k} \)
- \( \mathbf{F_2} = 3\hat{i} - 18\hat{j} - 4\hat{k} \)
- \( \mathbf{F_3} = a(\hat{i} - 3\hat{j} - \hat{k}) = a\hat{i} - 3a\hat{j} - a\hat{k} \)

2. **Write the Equation for Equilibrium**: For the particle to be in equilibrium, the sum of the forces must equal zero:
\[ \mathbf{F_1} + \mathbf{F_2} + \mathbf{F_3} = 0 \]

3. **Combine the Forces**: Substitute the forces into the equation:
\[ (2\hat{i} + 3\hat{j} - \hat{k}) + (3\hat{i} - 18\hat{j} - 4\hat{k}) + (a\hat{i} - 3a\hat{j} - a\hat{k}) = 0 \]

4. **Group the Components**: Combine the \( \hat{i} \), \( \hat{j} \), and \( \hat{k} \) components:
- \( \hat{i} \) components: \( 2 + 3 + a = 5 + a \)
- \( \hat{j} \) components: \( 3 - 18 - 3a = -15 - 3a \)
- \( \hat{k} \) components: \( -1 - 4 - a = -5 - a \)

Thus, we have:
\[ (5 + a)\hat{i} + (-15 - 3a)\hat{j} + (-5 - a)\hat{k} = 0 \]

5. **Set Each Component to Zero**: For the equation to hold true, each component must equal zero:
- \( 5 + a = 0 \)
- \( -15 - 3a = 0 \)
- \( -5 - a = 0 \)

6. **Solve for \( a \)**:
- From \( 5 + a = 0 \):
\[ a = -5 \]
- From \( -15 - 3a = 0 \):
\[ -3a = 15 \implies a = -5 \]
- From \( -5 - a = 0 \):
\[ -a = 5 \implies a = -5 \]

All three equations give the same result: \( a = -5 \).

### Final Answer:
The value of \( a \) is \( -5 \).