A force F = 2 i - λ j + 5 k is applied at the point A(1, 2, 5) . If...

The position vector of the point of application $A$ with respect to point $B$ is $\vec{r} = \vec{r}_A - \vec{r}_B$.

$\vec{r} = (1 - (-1))\hat{i} + (2 - (-2))\hat{j} + (5 - 3)\hat{k} = 2\hat{i} + 4\hat{j} + 2\hat{k}$

The moment of the force about point $B$ is given by $\vec{M} = \vec{r} \times \vec{F}$.

$\vec{M} = \begin{vmatrix} \hat{i} \text{ and } \hat{j} \text{ and } \hat{k} \\ 2 \text{ and } 4 \text{ and } 2 \\ 2 \text{ and } -\lambda \text{ and } 5 \end{vmatrix}$

$\vec{M} = \hat{i}(20 + 2\lambda) - \hat{j}(10 - 4) + \hat{k}(-2\lambda - 8)$

$\vec{M} = (20 + 2\lambda)\hat{i} - 6\hat{j} - (2\lambda + 8)\hat{k}$

It is given that $\vec{M} = 16\hat{i} - 6\hat{j} + 2\lambda\hat{k}$.

Comparing the coefficients of $\hat{i}$ and $\hat{k}$:

$20 + 2\lambda = 16 \Rightarrow 2\lambda = -4 \Rightarrow \lambda = -2$

$-2\lambda - 8 = 2\lambda \Rightarrow 4\lambda = -8 \Rightarrow \lambda = -2$

Both equations yield $\lambda = -2$.

Answer: $-2$