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?

Let P₁ : 2x + y $$-$$ z = 3 and P₂ : x + 2y + z = 2 be two planes. Then, which of the following statement(s) is(are) TRUE?

Let $$\widehat u = {u_1} \widehat i + {u_2}\widehat j + {u_3}\widehat k$$ be a unit vector in $${{R^3}}$$ and
$$\widehat w = {1 \over {\sqrt 6 }}\left( {\widehat i + \widehat j + 2\widehat k} \right).$$ Given that there exists a vector $${\overrightarrow v }$$ in $${{R^3}}$$ such that $$\left| {\widehat u \times \overrightarrow v } \right| = 1$$ and $$\widehat w.\left( {\widehat u \times \overrightarrow v } \right) = 1.$$ Which of the following statement(s) is (are) correct?

Consider a pyramid $$OPQRS$$ located in the first octant $$\left( {x \ge 0,y \ge 0,z \ge 0} \right)$$ with $$O$$ as origin, and $$OP$$ and $$OR$$ along the $$x$$-axis and the $$y$$-axis, respectively. The base $$OPQR$$ of the pyramid is a square with $$OP=3.$$ The point $$S$$ is directly above the mid-point, $$T$$ of diagonal $$OQ$$ such that $$TS=3.$$ Then