Let a line L passing through the point (1, 1, 1) be perpendicular to both the vectors 2i +…

Question

Let a line L passing through the point (1, 1, 1) be perpendicular to both the vectors $2\hat{i} + 2\hat{j} + \hat{k}$ and $\hat{i} + 2\hat{j} + 2\hat{k}$. If $P(a, b, c)$ is the foot of perpendicular from the origin on the line L, then the ...

Accepted Answer

Direction vector of line is

$=\begin{vmatrix} \hat{i} \quad  \hat{j} \quad  \hat{k} \ 1 \quad  2 \quad  2 \ 2 \quad  2 \quad  1 \end{vmatrix}=-2\hat{i}+3\hat{j}-2\hat{k}$

So equation of line is

$\frac{x-1}{2}=\frac{y-1}{-3}=\frac{z-1}{2}=\lambda $

let ( $2\lambda +1,-3\lambda +1,2\lambda +1$ ) be the foot of perpendicular from $(0,0,0)$ upon given line

$\begin{gathered} \Rightarrow \overset{	o \leftarrow }{OM}\cdot (2\hat{i}-3j+2\hat{k})=0 \ \Rightarrow 2(2\lambda +1)-3(-3\lambda +1)+2(2\lambda +1)=0 \ \Rightarrow \lambda =-\frac{1}{17} \end{gathered}$

So $(a,b,c)\equiv (\frac{-2}{17}+1,\frac{3}{17}+1,\frac{-2}{17}+1)$

$\Rightarrow 34(a+b+c)=100$

Solutions · 1