A particle moves 5m in the +x direction while being acted upon by a constant force`vecF = (4 N)hati + (2N)hatj - (4N)hatk` . The work done on the particle by this force is

Options

A. 20 J
B. 10 J
C. `-20 J`
D. 30 J

Explanation

To find the work done on the particle by the force, 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 Vector**:
The force vector is given as:
\[ \vec{F} = 4 \hat{i} + 2 \hat{j} - 4 \hat{k} \quad \text{(in Newtons)} \]

2. **Identify the Displacement Vector**:
The particle moves 5 meters in the +x direction. Thus, the displacement vector can be represented as:
\[ \vec{d} = 5 \hat{i} + 0 \hat{j} + 0 \hat{k} \quad \text{(in meters)} \]

3. **Calculate the Dot Product**:
The work done \( W \) is calculated using the dot product of the force vector and the displacement vector:
\[ W = \vec{F} \cdot \vec{d} \]
Expanding the dot product:
\[ W = (4 \hat{i} + 2 \hat{j} - 4 \hat{k}) \cdot (5 \hat{i} + 0 \hat{j} + 0 \hat{k}) \]
This simplifies to:
\[ W = (4 \cdot 5) + (2 \cdot 0) + (-4 \cdot 0) \]
\[ W = 20 + 0 + 0 = 20 \text{ Joules} \]

4. **Conclusion**:
The work done on the particle by the force is:
\[ W = 20 \text{ Joules} \]