Question

Find the value of λ such that the line $\frac{x-2}{6}=\frac{y-1}{\lambda }=\frac{z+5}{-4}$ is perpendicular to the plane 3x − y − 2z = 7.

Answer 1

$$\begin{gathered} \text{Direction ratios of the given line are proportional to 6, λ, -4.} \\ \mathrm{Direction\; ratios\; of\; the}\text{plane are 3, -1, -2.} \\ \text{Since the given line is parallel to the given plane, the line is perpendicular to the normal of the given plane.} \\ \Rightarrow \left(6\right)\left(3\right)+\left(\lambda \right)\left(-1\right)+\left(-4\right)\left(-2\right)=0 \\ \Rightarrow 18-\lambda +8=0 \\ \Rightarrow \lambda =26 \end{gathered}$$

Disclaimer: It should be "parallel" instead of "perpendicular" in the given problem.