The co-ordinates of the perpendicular drawn from the point $2 \hat{i}-\hat{j}+5 \hat{k}$ to the line…
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
$(1,-2,3)$
$(1,2,-3)$
$(-1,2,3)$
$(1,2,3)$
Solution
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
$\begin{aligned}
& (10 \lambda+9)(10)+(4 \lambda-1)(-4)+(-11 \lambda-13)(-11)=0 \\
& \therefore 100 \lambda+90+16 \lambda+4+121 \lambda+143=0 \Rightarrow 237=-237 \lambda \\
& \Rightarrow \lambda=-1 \\
& M=-10+11,4-2,11-8) \text { i.e. }(1,2,3)
\end{aligned}$