The co-ordinates of the perpendicular drawn from the point \(2 \hat{i}-\hat{j}+5 \hat{k}\) to the line \(\vec{r}=(11 \hat{i}-2 \hat{j}-8 \hat{k})+\lambda(10 \hat{i}-4 \hat{j}-11 \hat{k})\) are

We have \(\bar{r}=(11 \hat{i}-2 \hat{j}-8 \hat{k})+\lambda(10 \hat{i}-4 \hat{j}-11 \hat{k})\)
So coordinates of any point on this line are \([(10 \lambda+11),(-11 \lambda-2)\), \((-11 \lambda-8)]\)
Let \(\mathrm{P} \equiv(2,-1,5)\) and let \(\underline{\mathrm{M}}\) be foot of perpendicular.
\(\therefore\) d.r. of PM are \((10 \lambda+9),(-4 \lambda-1),(-11 \lambda-13)\)
Sibce \(\mathrm{PM}\) is perpendicular to given line, we write
\((10 \lambda+9)(10)+(4 \lambda-1)(-4)+(-11 \lambda-13)(-\)\(11)=0 \)
\( \therefore 100 \lambda+90+16 \lambda+4+121 \lambda+143=0 \Rightarrow 237=\) \(-237 \lambda \)
\( \Rightarrow \lambda=-1 \)
\( M=-10+11,4-2,11-8) \text { i.e. }(1,2,3)\)