Paragraph: A line L passes through the point P(5,-6,7) and is...

Let $a, b, c$ be the direction ratios of the line. Then its equation is $\frac{x-5}{a}=\frac{y+6}{b}=\frac{z-7}{c}$...(i) Since (i) is parallel to the planes $x+y+z=1 and 2 x-y-2 z=3$ then a(1)+b(1)+c(1)=0 \text{ and } a(2)+b(-1)+c(-2)=0 By cross multiplication $$\begin{aligned} & \frac{a}{-1}=\frac{b}{4}=\frac{c}{-3}=\lambda \\ & \Rightarrow a=-\lambda, b=4 \lambda, c=-3 \lambda \end{aligned}$$\Rightarrow$Direction ratios of the line are$ < -1,4,-3 > = < 1,-4,3 > $Substituting a,$\mathrm{b}, \mathrm{c}$in (i), we get$\frac{x-5}{-1}=\frac{y+6}{4}=\frac{z-7}{-3}$...(ii) Hence equation of the line is$\frac{x-5}{-1}=\frac{y+6}{4}=\frac{z-7}{-3}$