Let the position vectors of two points P and Q be 3 i - j +2 k and i +2 j -4 k, respectively. Let R and S be two points such that the direction…

Let the position vectors of two points P and Q be $\displaystyle 3 \hat{i}-\hat{j}+2 \hat{k}$ and $\displaystyle \hat{i}+2 \hat{j}-4 \hat{k}$, respectively. Let R and S be two points such that the direction ratios of lines PR and QS are ($\displaystyle 4$, -$\displaystyle 1$, $\displaystyle 2$) and (-$\displaystyle 2$, $\displaystyle 1$, -$\displaystyle 2$), respectively. Let lines PR and QS intersect at T. If the vector $\displaystyle \overrightarrow{T A}$ is perpendicular to both $\displaystyle \overrightarrow{P R}$ and $\displaystyle \overrightarrow{Q S}$ and the length of vector $\displaystyle \overrightarrow{T A}$ is $\displaystyle \sqrt{5}$ units, then the modulus of a position vector of A is :