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?

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

In $${R^3},$$ let $$L$$ be a straight lines passing through the origin. Suppose that all the points on $$L$$ are at a constant distance from the two planes $${P_1}:x + 2y - z + 1 = 0$$ and $${P_2}:2x - y + z - 1 = 0.$$ Let $$M$$ be the locus of the feet of the perpendiculars drawn from the points on $$L$$ to the plane $${P_1}.$$ Which of the following points lie (s) on $$M$$?