Volume of Parallelopiped

Q. If the volume of a parallelopiped formed by the vectors $\hat{i}+\lambda \hat{j}+\hat{k},\hat{j}+\lambda \hat{k}$ and $\lambda \hat{i}+\hat{k}$ is minimum, then $\lambda$ is equal to :

1. $-\frac{1}{\sqrt{3}}$
2. $-\sqrt{3}$
3. $\frac{1}{\sqrt{3}}$
4. $\sqrt{3}$

Q. Consider the cube in the first octant with sides $OP,OQ$ and $OR$ of length $1,$ along the x-axis, y-axis and z-axis, respectively, where $O(0,0,0)$ is the origin. Let $S(\frac{1}{2},\frac{1}{2},\frac{1}{2})$ be the centre of the cube and $T$ be the vertex of the cube opposite to the origin $O$ such that $S$ lies on the diagonal $OT.$If $\overrightarrow{p}=\overrightarrow{SP},\overrightarrow{q}=\overrightarrow{SQ},\overrightarrow{r}=\overrightarrow{SR}$ and $\overrightarrow{t}=\overrightarrow{ST},$ then the value of $|(\overrightarrow{p}\times \overrightarrow{q})\times (\overrightarrow{r}\times \overrightarrow{t})|$ is