Find the work done by the force `F=3 hat i- hat j-2 hat k` acrting on a particle such that the particle is displaced from point `A(-3,-4,1)topoin tB(-1,-1,-2)dot`

Explanation

To find the work done by the force \( \mathbf{F} = 3 \hat{i} - \hat{j} - 2 \hat{k} \) on a particle displaced from point \( A(-3, -4, 1) \) to point \( B(-1, -1, -2) \), we will follow these steps:

### Step 1: Calculate the Displacement Vector
The displacement vector \( \Delta \mathbf{s} \) from point \( A \) to point \( B \) can be calculated using the formula:
\[ \Delta \mathbf{s} = \mathbf{B} - \mathbf{A} \]
Where:
- \( \mathbf{A} = -3 \hat{i} - 4 \hat{j} + 1 \hat{k} \)
- \( \mathbf{B} = -1 \hat{i} - 1 \hat{j} - 2 \hat{k} \)

Calculating the components:
\[ \Delta \mathbf{s} = (-1 - (-3)) \hat{i} + (-1 - (-4)) \hat{j} + (-2 - 1) \hat{k} \]
\[ = (2) \hat{i} + (3) \hat{j} + (-3) \hat{k} \]
Thus, the displacement vector is:
\[ \Delta \mathbf{s} = 2 \hat{i} + 3 \hat{j} - 3 \hat{k} \]

### Step 2: Calculate the Work Done
The work done \( W \) by the force \( \mathbf{F} \) during the displacement \( \Delta \mathbf{s} \) is given by the dot product:
\[ W = \mathbf{F} \cdot \Delta \mathbf{s} \]
Substituting the values:
\[ \mathbf{F} = 3 \hat{i} - \hat{j} - 2 \hat{k} \]
\[ \Delta \mathbf{s} = 2 \hat{i} + 3 \hat{j} - 3 \hat{k} \]
Now, calculating the dot product:
\[ W = (3)(2) + (-1)(3) + (-2)(-3) \]
\[ = 6 - 3 + 6 \]
\[ = 9 \]

### Final Answer
The work done by the force is:
\[ W = 9 \]