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…

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 \(P R\) and \(Q S\) intersect at \(T\). If the vector \(\overrightarrow{T A}\) is perpendicular to both \(\overrightarrow{P R}\) and \(\overrightarrow{Q S}\) and the length of vector \(\overrightarrow{T A}\) is \(\sqrt{5}\) units, then the modulus of a position vector of \(A\) is:

[JEE Main 2021, 16 Mar (Shift 1)]