Three Dimensional Geometry: JEE Main 2021 Mathematics question

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