Thank you for reporting, we will resolve it shortly
Q.
The vector equation of the plane passing through the intersection of the planes $\vec{ r } .(\hat{ i }+\hat{ j }+\hat{ k })=1$ and $\vec{ r } .(\hat{ i }-2 \hat{ j })=-2,$ and the point (1,0,2) is :
$\vec{ r } \cdot(\hat{ i }+\hat{ j }+\hat{ k })=1$
$\vec{ r } \cdot(\hat{ i }-2 \hat{ j })=-2$
point (1,0,2) Eq $^{n}$ of plane
$\vec{ r } \cdot(\hat{ i }+\hat{ j }+\hat{ k })-1+\lambda\{ r .(\hat{ i }-2 \hat{ j })+2\}=0$
$\vec{ r } \cdot\{\hat{ i }(1+\lambda)+\hat{ j }(1-2 \lambda)+\hat{ k }(1)\}-1+2 \lambda=0$
Point $\hat{ i }+0 \hat{ j }+2 \hat{ k }=\vec{ r }$
$\therefore (\hat{ i }+2 \hat{ k }) \cdot\{\hat{ i }(1+\lambda)+\hat{ j }(1-2 \lambda)+\hat{ k }(1)\} -1+2 \lambda=0$
$1+\lambda+2-1+2 \lambda=0$
$\lambda=-\frac{2}{3}$
$\therefore \,\,\,\, \vec{ r } \cdot\left[\hat{ i }\left(\frac{1}{3}\right)+\hat{ j }\left(\frac{7}{3}\right)+\hat{ k}\right]$
$=\frac{7}{3}$
$r\cdot[\hat{ i }+7 \hat{ j }+3 \hat{ k }]=7$