A force `vec(F)=2hat(i)-3hat(j)+7hat(k) (N)` acts on a particle which undergoes a displacement `vec(r )=7hat(j)+3hat(j)-2hat(k)(m)`. Calculate the work done by the force.

Options

A. 37 J
B. `-9 J`
C. 49 J
D. 14 J

Answer

To calculate the work done by the force on the particle, we will 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 is given as:
\[ \vec{F} = 2\hat{i} - 3\hat{j} + 7\hat{k} \, \text{(N)} \]
- The displacement vector is given as:
\[ \vec{r} = 7\hat{i} + 3\hat{j} - 2\hat{k} \, \text{(m)} \]

2. **Calculate the Dot Product:**
The work done \( W \) is calculated using the dot product:
\[ W = \vec{F} \cdot \vec{r} = (2\hat{i} - 3\hat{j} + 7\hat{k}) \cdot (7\hat{i} + 3\hat{j} - 2\hat{k}) \]

3. **Perform the Dot Product Calculation:**
Using the property of dot products:
\[ \vec{A} \cdot \vec{B} = A_xB_x + A_yB_y + A_zB_z \]
where \( A_x, A_y, A_z \) are the components of vector \( \vec{A} \) and \( B_x, B_y, B_z \) are the components of vector \( \vec{B} \).

- Calculate each component:
- For \( \hat{i} \) components: \( 2 \times 7 = 14 \)
- For \( \hat{j} \) components: \( -3 \times 3 = -9 \)
- For \( \hat{k} \) components: \( 7 \times -2 = -14 \)

- Now, sum these results:
\[ W = 14 - 9 - 14 \]

4. **Final Calculation of Work Done:**
\[ W = 14 - 9 - 14 = -9 \, \text{Joules} \]

### Conclusion:
The work done by the force on the particle is:
\[ W = -9 \, \text{Joules} \]