A particle is displaced from `A=(2,2,4)` to `B=(5, -3,-1)` . A constant force of `34N` acts in the direction of `vec(AP)`. Where `P=(10,2,-11)`. (Coordinates are in m).
(i) Find the `(vec(F))` .
(ii) Find the work done by the force to cause a displacement.

Answer

To solve the problem step by step, we will break it down into two parts: finding the force vector \( \vec{F} \) and then calculating the work done by this force during the displacement.

### Step 1: Finding the Force Vector \( \vec{F} \)

1. **Identify the Points**:
- Point A: \( A = (2, 2, 4) \)
- Point B: \( B = (5, -3, -1) \)
- Point P: \( P = (10, 2, -11) \)

2. **Calculate the Vector \( \vec{AP} \)**:
\[ \vec{AP} = \vec{P} - \vec{A} = (10 - 2, 2 - 2, -11 - 4) = (8, 0, -15) \]
Thus, \( \vec{AP} = 8\hat{i} + 0\hat{j} - 15\hat{k} \).

3. **Find the Magnitude of \( \vec{AP} \)**:
\[ |\vec{AP}| = \sqrt{(8)^2 + (0)^2 + (-15)^2} = \sqrt{64 + 0 + 225} = \sqrt{289} = 17 \]

4. **Find the Unit Vector \( \hat{AP} \)**:
\[ \hat{AP} = \frac{\vec{AP}}{|\vec{AP}|} = \frac{8\hat{i} + 0\hat{j} - 15\hat{k}}{17} = \left(\frac{8}{17}\hat{i} + 0\hat{j} - \frac{15}{17}\hat{k}\right) \]

5. **Calculate the Force Vector \( \vec{F} \)**:
The force \( \vec{F} \) has a magnitude of \( 34 \, \text{N} \) and acts in the direction of \( \hat{AP} \):
\[ \vec{F} = |\vec{F}| \hat{AP} = 34 \left(\frac{8}{17}\hat{i} + 0\hat{j} - \frac{15}{17}\hat{k}\right) = 16\hat{i} + 0\hat{j} - 30\hat{k} \]
Thus, \( \vec{F} = 16\hat{i} - 30\hat{k} \).

### Step 2: Finding the Work Done by the Force

1. **Calculate the Displacement Vector \( \vec{AB} \)**:
\[ \vec{AB} = \vec{B} - \vec{A} = (5 - 2, -3 - 2, -1 - 4) = (3, -5, -5) \]
Thus, \( \vec{AB} = 3\hat{i} - 5\hat{j} - 5\hat{k} \).

2. **Calculate the Work Done \( W \)**:
The work done by the force is given by the dot product of \( \vec{F} \) and \( \vec{AB} \):
\[ W = \vec{F} \cdot \vec{AB} = (16\hat{i} + 0\hat{j} - 30\hat{k}) \cdot (3\hat{i} - 5\hat{j} - 5\hat{k}) \]
\[ W = 16 \cdot 3 + 0 \cdot (-5) + (-30) \cdot (-5) = 48 + 0 + 150 = 198 \, \text{J} \]

### Final Answers:
- (i) The force vector \( \vec{F} = 16\hat{i} - 30\hat{k} \).
- (ii) The work done by the force is \( 198 \, \text{J} \).