Question

Find the value of λ so that the following lines are perpendicular to each other. $\frac{x-5}{5\lambda +2}=\frac{2-y}{5}=\frac{1-z}{-1},\frac{x}{1}=\frac{2y+1}{4\lambda }=\frac{1-z}{-3}$

Answer 1

The equations of the given lines $\frac{x-5}{5\lambda +2}=\frac{2-y}{5}=\frac{1-z}{-1}\mathrm{and}\frac{x}{1}=\frac{2y+1}{4\lambda }=\frac{1-z}{-3}$ can be re-written as

$\frac{x-5}{5\lambda +2}=\frac{y-2}{-5}=\frac{z-1}{1}\mathrm{and}\frac{x}{1}=\frac{y+{\displaystyle \frac{1}{2}}}{2\lambda }=\frac{z-1}{3}$


Since the given lines are perpendicular to each other, we have

$$\begin{gathered} \left(5\lambda +2\right)1-5\left(2\lambda \right)+1\left(3\right)=0 \\ \Rightarrow 5\lambda =5 \\ \Rightarrow \lambda =1 \end{gathered}$$