Given vectors

Force: $\vec F = 2\hat i + b\hat j + \hat k$

Displacement: $\vec s = \hat i - 2\hat j - \hat k$

Condition for zero work

Work done is given by the dot product: $W = \vec F \cdot \vec s$

Since work done is zero, $\vec F \cdot \vec s = 0$

Compute the dot product

$\vec F \cdot \vec s = (2)(1) + (b)(-2) + (1)(-1)$

$= 2 - 2b - 1$

$= 1 - 2b$

Set equal to zero

$1 - 2b = 0$

$2b = 1$

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

Match with options

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

So the correct option is B.