Question

The lines $\frac{x+3}{-2}=\frac{y}{1}=\frac{z-4}{3}$ and $\frac{x}{\lambda }=\frac{y-1}{\lambda +1}=\frac{z}{\lambda +2}$ are perpendicular to each other. Then $\lambda$ is equal to

• A. $-\frac{7}{2}$
• B. $4$
• C. $-\frac{1}{4}$
• D. $-4$

Answer 1

Answer: (A)

$-\frac{7}{2}$

The lines $\frac{x+3}{-2}=\frac{y}{1}=\frac{z-4}{3}$ and $\frac{x}{\lambda }=\frac{y-1}{\lambda +1}=\frac{z}{\lambda +2}$

The direction vector of the first line is $-2i+j+3k$ and second line $\lambda i+(\lambda +1)j+(\lambda +2)k$.

Since they are perpendicular, dot product is zero.

$\Rightarrow$ $-2\lambda +\lambda +1+3\lambda +6=0$

$\Rightarrow \lambda =\frac{-7}{2}$.