A force $$\overrightarrow{F}=2\widehat{i}+b\widehat{j}+\widehat{k}$$ is applied on a particle and it undergoes a displacement $$\widehat{i}-2\widehat{j}-\widehat{k}$$.What will be the value of , if work done on the particle is zero.

We need to find the value of $$b$$ such that the work done is zero when force $$\vec{F} = 2\hat{i} + b\hat{j} + \hat{k}$$ causes displacement $$\vec{d} = \hat{i} - 2\hat{j} - \hat{k}$$.

Apply the work formula

Work done = $$\vec{F} \cdot \vec{d} = 0$$

Compute the dot product

$$W = (2)(1) + (b)(-2) + (1)(-1) = 2 - 2b - 1 = 1 - 2b$$

Set work equal to zero and solve

$$1 - 2b = 0$$

$$b = \frac{1}{2}$$

The answer is Option B: $$\frac{1}{2}$$.