The work done by a force `vec F = 3 hat i -4hat j +5hat k` displaces the body from a point `(3,4,6)` to a point `(7,2,5)` is

Options

A. 15 units
B. 25 units
C. 20 units
D. 10 units

Explanation

To find the work done by the force \(\vec{F} = 3\hat{i} - 4\hat{j} + 5\hat{k}\) during the displacement from the point \((3, 4, 6)\) to the point \((7, 2, 5)\), we can follow these steps:

### Step 1: Identify the Initial and Final Points
The initial point \( \vec{S}_{initial} \) is given as:
\[ \vec{S}_{initial} = 3\hat{i} + 4\hat{j} + 6\hat{k} \]
The final point \( \vec{S}_{final} \) is given as:
\[ \vec{S}_{final} = 7\hat{i} + 2\hat{j} + 5\hat{k} \]

### Step 2: Calculate the Displacement Vector
The displacement vector \( \vec{S} \) can be calculated as:
\[ \vec{S} = \vec{S}_{final} - \vec{S}_{initial} \]
Calculating this gives:
\[ \vec{S} = (7\hat{i} + 2\hat{j} + 5\hat{k}) - (3\hat{i} + 4\hat{j} + 6\hat{k}) \]
\[ \vec{S} = (7 - 3)\hat{i} + (2 - 4)\hat{j} + (5 - 6)\hat{k} \]
\[ \vec{S} = 4\hat{i} - 2\hat{j} - 1\hat{k} \]

### Step 3: Calculate the Work Done
The work done \( W \) by the force is given by the dot product of the force vector \( \vec{F} \) and the displacement vector \( \vec{S} \):
\[ W = \vec{F} \cdot \vec{S} \]
Substituting the values:
\[ W = (3\hat{i} - 4\hat{j} + 5\hat{k}) \cdot (4\hat{i} - 2\hat{j} - 1\hat{k}) \]
Calculating the dot product:
\[ W = (3 \cdot 4) + (-4 \cdot -2) + (5 \cdot -1) \]
\[ W = 12 + 8 - 5 \]
\[ W = 15 \]

### Conclusion
The work done by the force is:
\[ \boxed{15 \text{ joules}} \]