If a force of `(8.0 hati+2.0 hatj+3 hatk)N` displaces a body through a distance of `(6.0 hati-3 .0 hatj -9 hatk)`m . The work done will be

Options

A. 15 J
B. 34 J
C. 56 J
D. 42 J

Explanation

To find the work done by a force when a body is displaced, 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 Force and Displacement Vectors:**
- The force vector \( \mathbf{F} \) is given as:
\[ \mathbf{F} = 8 \hat{i} + 2 \hat{j} + 3 \hat{k} \, \text{N} \]
- The displacement vector \( \mathbf{d} \) is given as:
\[ \mathbf{d} = 6 \hat{i} - 3 \hat{j} - 9 \hat{k} \, \text{m} \]

2. **Calculate the Dot Product:**
- The work done \( W \) is calculated using the dot product:
\[ W = \mathbf{F} \cdot \mathbf{d} \]
- The dot product is calculated as follows:
\[ W = (8 \hat{i} + 2 \hat{j} + 3 \hat{k}) \cdot (6 \hat{i} - 3 \hat{j} - 9 \hat{k}) \]

3. **Expand the Dot Product:**
- Using the properties of dot product:
\[ W = (8 \cdot 6) + (2 \cdot -3) + (3 \cdot -9) \]
- Calculate each term:
- \( 8 \cdot 6 = 48 \)
- \( 2 \cdot -3 = -6 \)
- \( 3 \cdot -9 = -27 \)

4. **Sum the Results:**
- Combine the results:
\[ W = 48 - 6 - 27 \]
- Calculate:
\[ W = 48 - 6 = 42 \]
\[ W = 42 - 27 = 15 \, \text{J} \]

5. **Final Answer:**
- The work done is:
\[ W = 15 \, \text{J} \]