Let a and b be positive real numbers. Suppose $$PQ = a\widehat i + b\widehat j$$ and $$PS = a\widehat i - b\widehat j$$ are adjacent sides of a parallelogram PQRS. Let u and v be the projection vectors of $$w = \widehat i + \widehat j$$ along PQ and PS, respectively. If |u| + |v| = |w| and if the area of the parallelogram PQRS is 8, then which of the following statements is/are TRUE?

Let L₁ and L₂ be the following straight lines.

$${L_1}:{{x - 1} \over 1} = {y \over { - 1}} = {{z - 1} \over 3}$$ and $${L_2}:{{x - 1} \over { - 3}} = {y \over { - 1}} = {{z - 1} \over 1}$$.

Suppose the straight line

$$L:{{x - \alpha } \over l} = {{y - 1} \over m} = {{z - \gamma } \over { - 2}}$$

lies in the plane containing L₁ and L₂ and passes through the point of intersection of L₁ and L₂. If the line L bisects the acute angle between the lines L₁ and L₂, then which of the following statements is/are TRUE?

Three lines $${L_1}:r = \lambda \widehat i$$, $$\lambda $$ $$ \in $$ R,

$${L_2}:r = \widehat k + \mu \widehat j$$, $$\mu $$ $$ \in $$ R and

$${L_3}:r = \widehat i + \widehat j + v\widehat k$$, v $$ \in $$ R are given.

For which point(s) Q on L₂ can we find a point P on L₁ and a point R on L₃ so that P, Q and R are collinear?

Let L₁ and L₂ denote the lines

$$r = \widehat i + \lambda ( - \widehat i + 2\widehat j + 2\widehat k)$$, $$\lambda $$$$ \in $$ R

and $$r = \mu (2\widehat i - \widehat j + 2\widehat k),\,\mu \in R$$

respectively. If L₃ is a line which is perpendicular to both L₁ and L₂ and cuts both of them, then which of the following options describe(s) L₃?