A force `vec F = (5 vec i+ 3 vec j+ 2 vec k) N` is applied over a particle which displaces it from its origin to the point `vec r=(2 vec i- vec j) m`. The work done on the particle in joules is.

Options

A. `+ 10`
B. ` + 7`
C. `-7`
D. `+ 13`

Explanation

To find the work done on a particle when a force is applied, we can use the formula for work done in vector form, which is given by the dot product of the force vector and the displacement vector.

### Step-by-step Solution:

1. **Identify the Force Vector and Displacement Vector:**
- The force vector is given as \( \vec{F} = 5 \hat{i} + 3 \hat{j} + 2 \hat{k} \) N.
- The displacement vector is given as \( \vec{r} = 2 \hat{i} - 1 \hat{j} \) m. Since the particle moves from the origin, the displacement vector \( \vec{s} \) is equal to \( \vec{r} \).

2. **Write the Work Done Formula:**
- The work done \( W \) is calculated using the dot product:
\[ W = \vec{F} \cdot \vec{s} \]

3. **Calculate the Dot Product:**
- The dot product \( \vec{F} \cdot \vec{s} \) is calculated as follows:
\[ \vec{F} \cdot \vec{s} = (5 \hat{i} + 3 \hat{j} + 2 \hat{k}) \cdot (2 \hat{i} - 1 \hat{j} + 0 \hat{k}) \]
- Using the properties of the dot product:
\[ = (5 \cdot 2) + (3 \cdot -1) + (2 \cdot 0) \]
- Calculate each term:
\[ = 10 - 3 + 0 \]
- Combine the results:
\[ = 7 \text{ J} \]

4. **Final Result:**
- The work done on the particle is \( 7 \) joules.