A force `F = 2i+j-k` acts as a point A whose position vector is `2i-j`. IF point of application of F moves from the point A to the point B with P.V. `2i+j`, then the work done by F is

Options

A. 4
B. 20
C. 2
D. none

Answer

To solve the problem step by step, we will calculate the work done by the force \( F \) as it moves from point \( A \) to point \( B \).

### Step 1: Identify the given vectors
The force vector \( F \) is given as:
\[ F = 2\mathbf{i} + \mathbf{j} - \mathbf{k} \]

The position vector of point \( A \) is:
\[ \mathbf{A} = 2\mathbf{i} - \mathbf{j} \]

The position vector of point \( B \) is:
\[ \mathbf{B} = 2\mathbf{i} + \mathbf{j} \]

### Step 2: Calculate the displacement vector \( \mathbf{AB} \)
The displacement vector \( \mathbf{AB} \) can be calculated as:
\[ \mathbf{AB} = \mathbf{B} - \mathbf{A} \]
Substituting the values:
\[ \mathbf{AB} = (2\mathbf{i} + \mathbf{j}) - (2\mathbf{i} - \mathbf{j}) \]
\[ = 2\mathbf{i} + \mathbf{j} - 2\mathbf{i} + \mathbf{j} \]
\[ = 2\mathbf{j} \]

### Step 3: Calculate the work done using the formula
The work done \( W \) by the force when moving from point \( A \) to point \( B \) is given by the dot product of the force vector and the displacement vector:
\[ W = \mathbf{F} \cdot \mathbf{AB} \]
Substituting the vectors:
\[ W = (2\mathbf{i} + \mathbf{j} - \mathbf{k}) \cdot (2\mathbf{j}) \]

### Step 4: Calculate the dot product
Calculating the dot product:
\[ W = 2\mathbf{i} \cdot 2\mathbf{j} + \mathbf{j} \cdot 2\mathbf{j} - \mathbf{k} \cdot 2\mathbf{j} \]
Since \( \mathbf{i} \cdot \mathbf{j} = 0 \) and \( \mathbf{k} \cdot \mathbf{j} = 0 \):
\[ W = 0 + 2 + 0 = 2 \]

### Final Answer
The work done by the force \( F \) is:
\[ W = 2 \text{ units} \]