Mathematics - Vector Question with Solution | TestHub

Line 1 :$\frac{x}{1}=\frac{y}{1}=\frac{z-1}{0}-r, Q\left(r, r, 1\right)$
Line 2 :$\frac{x}{1}=\frac{y}{-1}=\frac{z+1}{0}=k, R\left(k, -k, -1\right)$
$\overrightarrow{PQ}=\left(\lambda -r\right) \widehat{i}+\left(\lambda -r\right) \widehat{j}+\left(\lambda -1\right) \widehat{k}$
And$\lambda -r+\lambda -r=0$as$\overrightarrow{PQ}$is$\perp$to$L_1$
$\Rightarrow$$2\lambda =2r \Rightarrow \lambda =r$
$\overrightarrow{PR}=\left(\lambda -k\right)\widehat{i}+\left(\lambda +k\right) \widehat{j}+\left(\lambda +1\right) \widehat{k}$
And$\lambda -k-\lambda -k=0$as$\overrightarrow{PR}$is$\perp$to$L_2$
$\Rightarrow$$k=0$
So$PQ\perp PR$
$\left(\lambda -r\right) \left(\lambda -k\right)+\left(\lambda -r\right) \left(\lambda +k\right)+\left(\lambda -1\right) \left(\lambda +1\right)=0$
$\Rightarrow$$\lambda =1, -1$
For$\lambda =1$as point P and Q coincide
$\Rightarrow$$\lambda = -1.$