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

<p>To determine the value of $b$ such that the work done on the particle is zero, follow these steps:</p> <p><p>The work done by a force on a displacement is given by the dot product:</p> <p>$\text{Work} = \overrightarrow{F} \cdot \overrightarrow{d}.$</p></p> <p><p>Given the force </p> <p>$\overrightarrow{F} = 2\hat{i} + b\hat{j} + \hat{k},$ </p> <p>and the displacement </p> <p>$\overrightarrow{d} = \hat{i} - 2\hat{j} - \hat{k},$ </p> <p>compute the dot product:</p> <p>$\overrightarrow{F} \cdot \overrightarrow{d} = (2)(1) + (b)(-2) + (1)(-1).$</p></p> <p><p>Simplify the expression:</p> <p>$\overrightarrow{F} \cdot \overrightarrow{d} = 2 - 2b - 1 = 1 - 2b.$</p></p> <p><p>Since the work done is zero:</p> <p>$1 - 2b = 0.$</p></p> <p><p>Solve for $b$:</p> <p>$2b = 1,$</p> <p>$b = \frac{1}{2}.$</p></p> <p>Thus, the value of $b$ is $\frac{1}{2}$, which corresponds to Option B.</p>