A force `vecF=6hati+2hatj-3hatk` acts on a particle and produces a displacement of `vecs=2hati-3hatj+xhatk`. If the work done is zero, the value of x is

Options

A. `-2`
B. `1//2`
C. `6`
D. `2`

Answer

To find the value of \( x \) such that the work done by the force \( \vec{F} = 6\hat{i} + 2\hat{j} - 3\hat{k} \) on a particle that undergoes a displacement \( \vec{s} = 2\hat{i} - 3\hat{j} + x\hat{k} \) is zero, we can follow these steps:

### Step 1: Understand the Work Done Formula
The work done \( W \) by a force \( \vec{F} \) during a displacement \( \vec{s} \) is given by the dot product:
\[ W = \vec{F} \cdot \vec{s} \]

### Step 2: Write the Force and Displacement Vectors
We have:
\[ \vec{F} = 6\hat{i} + 2\hat{j} - 3\hat{k} \]
\[ \vec{s} = 2\hat{i} - 3\hat{j} + x\hat{k} \]

### Step 3: Calculate the Dot Product
Now, we calculate the dot product \( \vec{F} \cdot \vec{s} \):
\[ \vec{F} \cdot \vec{s} = (6\hat{i} + 2\hat{j} - 3\hat{k}) \cdot (2\hat{i} - 3\hat{j} + x\hat{k}) \]
Using the properties of dot product, we get:
\[ \vec{F} \cdot \vec{s} = 6 \cdot 2 + 2 \cdot (-3) + (-3) \cdot x \]
\[ = 12 - 6 - 3x \]
\[ = 6 - 3x \]

### Step 4: Set the Work Done to Zero
According to the problem, the work done is zero:
\[ 6 - 3x = 0 \]

### Step 5: Solve for \( x \)
Now, we solve for \( x \):
\[ 6 = 3x \]
\[ x = \frac{6}{3} = 2 \]

### Conclusion
Thus, the value of \( x \) is \( 2 \).