The length of the perpendicular from the point 1 , - 2 , 5 on the line passing through 1 , 2 , 4 and…

Given line is parallel to the line $x+y-z=0=x-2y+3z-5$ is:

So, direction ratio of the line $=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 1& 1& -1\\ 1& -2& 3\end{array}\right|=\hat{i}-4\hat{j}-3\hat{k}$

So, equation of line passing through $\left(1,2,4\right)$ is

$\overrightarrow{r}=\hat{i}+2\hat{j}+4\hat{k}+\lambda \left(\hat{i}-4\hat{j}-3\hat{k}\right)$

Let $A\left(1,2,4\right)$,$Q\left(1,-2,5\right)$ and $P$ be $\left(1+\lambda ,2-4\lambda ,4-3\lambda \right)$

Given $\overrightarrow{PQ}$ and line are perpendicular,

So, $\overrightarrow{PQ}·\left(\hat{i}-4\hat{j}-3\hat{k}\right)=0$

$\Rightarrow \left(\lambda \hat{i}+\left(4-4\lambda \right)\hat{j}-\left(1+3\lambda \right)\hat{k}\right).\left(\hat{i}-4\hat{j}-3\hat{k}\right)=0$

$\Rightarrow \lambda -16+16\lambda +3+9\lambda =0$

$\Rightarrow \lambda =\frac{13}{26}=\frac{1}{2}$

Hence $\overrightarrow{PQ}\equiv \left(\frac{1}{2},2,\frac{-5}{2}\right)$

Now $\left|PQ\right|=\sqrt{\frac{21}{2}}$