The work done by the forces `vecF = 2hati - hatj -hatk` in moving an object along the vectors `3hati + 2hatj - 5hatk` is:

Options

A. `-9` units
B. 15 units
C. 9 units
D. None of these

Answer

To find the work done by the force \(\vec{F} = 2\hat{i} - \hat{j} - \hat{k}\) while moving an object along the vector \(\vec{r} = 3\hat{i} + 2\hat{j} - 5\hat{k}\), we can use the formula for work done, which is given by the dot product of the force vector and the displacement vector.

### Step-by-Step Solution:

1. **Identify the vectors**:
- Force vector: \(\vec{F} = 2\hat{i} - \hat{j} - \hat{k}\)
- Displacement vector: \(\vec{r} = 3\hat{i} + 2\hat{j} - 5\hat{k}\)

2. **Apply the dot product formula**:
The work done \(W\) is calculated using the dot product:
\[ W = \vec{F} \cdot \vec{r} \]

3. **Calculate the dot product**:
\[ W = (2\hat{i} - \hat{j} - \hat{k}) \cdot (3\hat{i} + 2\hat{j} - 5\hat{k}) \]
The dot product is computed as follows:
\[ W = (2 \cdot 3) + (-1 \cdot 2) + (-1 \cdot -5) \]
Breaking it down:
- \(2 \cdot 3 = 6\)
- \(-1 \cdot 2 = -2\)
- \(-1 \cdot -5 = 5\)

4. **Combine the results**:
\[ W = 6 - 2 + 5 \]
\[ W = 4 + 5 = 9 \]

5. **Final result**:
The work done by the force is:
\[ W = 9 \text{ units} \]