Let P Q R be a triangle such that P Q =-2 i - j +2 k and PR =a i +b j -4 k, a, b ∈ Z. Let S be the point on QR, which is equidistant from the lines…

Let $\displaystyle P Q R$ be a triangle such that $\displaystyle \overrightarrow{P Q}=-2 \hat{i}-\hat{j}+2 \hat{k}$ and $\displaystyle \overrightarrow{\mathrm{PR}}=a \hat{\mathrm{i}}+b \hat{\mathrm{j}}-4 \hat{\mathrm{k}}, a, b \in \mathbb{Z}$. Let S be the point on QR , which is equidistant from the lines PQ and PR . If $\displaystyle |\overrightarrow{\mathrm{PR}}|=9$ and $\displaystyle \overrightarrow{\mathrm{PS}}=\hat{\mathrm{i}}-7 \hat{\mathrm{j}}+2 \hat{\mathrm{k}}$, then the value of $\displaystyle 3 a-4 b$ is $\displaystyle \_\_\_\_$