A body constrained to move along y-axis is subjected to a constant force `vecF= -hati+2hatj+3hatk N`. The work done by this force in moving the body a distance of 4 m along y-axis is

Options

A. 8 J
B. 4 J
C. 2 J
D. 6 J

Answer

To solve the problem, we need to calculate the work done by the force vector when the body moves a distance of 4 meters along the y-axis. Here’s a step-by-step solution:

### Step 1: Identify the Force Vector
The force vector is given as:
\[ \vec{F} = -\hat{i} + 2\hat{j} + 3\hat{k} \, \text{N} \]

### Step 2: Determine the Displacement Vector
Since the body is moving along the y-axis for a distance of 4 meters, the displacement vector can be represented as:
\[ \vec{d} = 4\hat{j} \, \text{m} \]

### Step 3: Calculate the Work Done
The work done \( W \) by a force is calculated using the dot product of the force vector and the displacement vector:
\[ W = \vec{F} \cdot \vec{d} \]
Substituting the values of \( \vec{F} \) and \( \vec{d} \):
\[ W = (-\hat{i} + 2\hat{j} + 3\hat{k}) \cdot (4\hat{j}) \]

### Step 4: Perform the Dot Product
Using the properties of the dot product:
- The dot product of two vectors is zero if they are perpendicular (i.e., different unit vectors).
- The dot product of a unit vector with itself is 1.

Calculating the dot product:
\[ W = (-\hat{i} \cdot 4\hat{j}) + (2\hat{j} \cdot 4\hat{j}) + (3\hat{k} \cdot 4\hat{j}) \]
Since \(\hat{i} \cdot \hat{j} = 0\) and \(\hat{k} \cdot \hat{j} = 0\), we have:
\[ W = 0 + 8 + 0 = 8 \, \text{J} \]

### Conclusion
The work done by the force in moving the body a distance of 4 meters along the y-axis is:
\[ \boxed{8 \, \text{J}} \]