The work done in moving an object from origin to a point whose position vector is `vecr = 3hati + 2hatj - 5hatk` by a force `vecF = 2hati - hatj - hatk` is

Options

A. 1 unit
B. 9 unit
C. 13 unit
D. 60 unit

Answer

To solve the problem of calculating the work done in moving an object from the origin to a point defined by the position vector \(\vec{r} = 3\hat{i} + 2\hat{j} - 5\hat{k}\) under the influence of a force \(\vec{F} = 2\hat{i} - \hat{j} - \hat{k}\), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Displacement Vector**:
The displacement vector \(\vec{s}\) is the same as the position vector \(\vec{r}\) since the object is moving from the origin (0,0,0) to the point (3,2,-5).
\[ \vec{s} = \vec{r} = 3\hat{i} + 2\hat{j} - 5\hat{k} \]

2. **Write the Formula for Work Done**:
The work done \(W\) by a force \(\vec{F}\) when moving an object through a displacement \(\vec{s}\) is given by the dot product:
\[ W = \vec{F} \cdot \vec{s} \]

3. **Substitute the Force and Displacement Vectors**:
Substitute the given vectors into the formula:
\[ \vec{F} = 2\hat{i} - \hat{j} - \hat{k} \]
\[ \vec{s} = 3\hat{i} + 2\hat{j} - 5\hat{k} \]

4. **Calculate the Dot Product**:
The dot product \(\vec{F} \cdot \vec{s}\) is calculated as follows:
\[ \vec{F} \cdot \vec{s} = (2\hat{i} - \hat{j} - \hat{k}) \cdot (3\hat{i} + 2\hat{j} - 5\hat{k}) \]
Using the properties of dot product:
\[ = 2 \cdot 3 + (-1) \cdot 2 + (-1) \cdot (-5) \]
\[ = 6 - 2 + 5 \]
\[ = 9 \]

5. **Conclusion**:
The work done in moving the object is:
\[ W = 9 \text{ units} \]

### Final Answer:
The work done is \(9\) units.