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. **Write the Dot Product Formula:**
- The work done \( W \) by the force is calculated using the dot product:
\[ W = \vec{F} \cdot \vec{r} \]

3. **Calculate the Dot Product:**
- The dot product of two vectors \( \vec{A} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k} \) and \( \vec{B} = b_1\hat{i} + b_2\hat{j} + b_3\hat{k} \) is given by:
\[ \vec{A} \cdot \vec{B} = a_1b_1 + a_2b_2 + a_3b_3 \]
- Applying this to our vectors:
\[ W = (2)(7) + (-3)(3) + (7)(-2) \]

4. **Perform the Multiplications:**
- Calculate each term:
- \( 2 \times 7 = 14 \)
- \( -3 \times 3 = -9 \)
- \( 7 \times -2 = -14 \)

5. **Sum the Results:**
- Now, sum the results of the multiplications:
\[ W = 14 - 9 - 14 \]

6. **Calculate the Final Work Done:**
- Simplifying the expression:
\[ W = 14 - 9 - 14 = -9 \, \text{Joules} \]

### Final Answer:
The work done by the force is \( W = -9 \, \text{J} \).