Three forces `vecF_(1)=(3hati+2hatj-hatk)` dynes, `vecF_(2)=(3hati+4hatj-5hatk)` dynes and `vecF_(3)=a(hati+hatj-hatk)` dynes act simultaneously on a particle. What should be the value of 'a' so that the particle may remain in equilibrium?

Explanation

To find the value of 'a' such that the particle remains in equilibrium under the influence of the three forces, we need to ensure that the net force acting on the particle is zero. This can be achieved by summing the forces and setting their components equal to zero.

### Step-by-Step Solution:

1. **Write down the forces:**
- \( \vec{F_1} = 3\hat{i} + 2\hat{j} - \hat{k} \)
- \( \vec{F_2} = 3\hat{i} + 4\hat{j} - 5\hat{k} \)
- \( \vec{F_3} = a(\hat{i} + \hat{j} - \hat{k}) = a\hat{i} + a\hat{j} - a\hat{k} \)
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