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