Question

If the lines $\frac{x-1}{-3}=\frac{y-2}{2\lambda }=\frac{z-3}{2}\mathrm{and}\frac{x-1}{3\lambda }=\frac{y-1}{1}=\frac{z-6}{-5}$ are perpendicular, find the value of λ.

Answer 1

The equations of the given lines are

$$\begin{gathered} \frac{x-1}{-3}=\frac{y-2}{2\lambda }=\frac{z-3}{2} \\ \frac{x-1}{3\lambda }=\frac{y-1}{1}=\frac{z-6}{-5} \end{gathered}$$

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

$$\begin{gathered} -3\left(3\lambda \right)+2\lambda \left(1\right)+2\left(-5\right)=0 \\ \Rightarrow -9\lambda +2\lambda -10=0 \\ \Rightarrow \lambda =-\frac{10}{7} \end{gathered}$$