If `vec F =2 hat i + 3hat j +4hat k` acts on a body and displaces it by `vec S =3 hat i + 2hat j + 5 hat k`, then the work done by the force is

Options

A. 12 J
B. 20 J
C. 32 J
D. 64 J

Explanation

To find the work done by the force \(\vec{F}\) on the body during the displacement \(\vec{S}\), we can use the formula for work done, which is given by the dot product of the force vector and the displacement vector:

\[ W = \vec{F} \cdot \vec{S} \]

### Step 1: Identify the vectors
Given:
\[ \vec{F} = 2\hat{i} + 3\hat{j} + 4\hat{k} \]
\[ \vec{S} = 3\hat{i} + 2\hat{j} + 5\hat{k} \]

### Step 2: Calculate the dot product
The dot product \(\vec{F} \cdot \vec{S}\) is calculated as follows:

\[ \vec{F} \cdot \vec{S} = (2\hat{i} + 3\hat{j} + 4\hat{k}) \cdot (3\hat{i} + 2\hat{j} + 5\hat{k}) \]

Using the properties of the dot product, we can expand this:

\[ = 2 \cdot 3 + 3 \cdot 2 + 4 \cdot 5 \]

### Step 3: Perform the multiplication
Calculating each term:

\[ = 6 + 6 + 20 \]

### Step 4: Sum the results
Now, add the results together:

\[ = 6 + 6 + 20 = 32 \]

### Conclusion
Thus, the work done by the force is:

\[ W = 32 \, \text{Joules} \]

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