Question

Let $L _1: \frac{ x -1}{1}=\frac{ y -1}{2}=\frac{ z -1}{ k }$ and $L _2: \frac{ x -3}{1}=\frac{ y -2}{-1}=\frac{ z -3}{-1}$ be two intersecting lines and $L _3$ be a straight line which is perpendicular to $L _1$ and $L _2$ and passing through their point of intersection. If length of perpendicular from $(3,1,2)$ to the line $L _3$ is $\lambda$ then $[\lambda]$ equals
[Note: [y] denotes the greatest integer function less than or equal to $y$.]

Answer 1

$ L _1$ and $L _2$ are intersecting $\Rightarrow k =3$ and their point of intersection is $(2,3,4)$.
$L_3: \frac{x-2}{1}=\frac{y-3}{4}=\frac{z-4}{-3} $
$P=\frac{1(\hat{i}-2 \hat{j}-2 \hat{k}) \times(\hat{i}+4 \hat{j}-3 \hat{k})}{|\hat{i}+4 \hat{j}-3 \hat{k}|}=\frac{\sqrt{233}}{\sqrt{26}}$

${[P]=[\sqrt{8 \cdot 9}=2 .}$