Consider the lines L_1: x-12=y-31=z-22 and L_2: x-21=y-22=z-33.

Question

Consider the lines $L_1:\ \dfrac{x-1}{2}=\dfrac{y-3}{1}=\dfrac{z-2}{2}$ and $L_2:\ \dfrac{x-2}{1}=\dfrac{y-2}{2}=\dfrac{z-3}{3}$. A line $L_3$ having direction ratios $1,-1,-2$ intersects $L_1$ and $L_2$ at points $P$ and $Q$ respectively. Then the length of segment $PQ$ is:

Accepted Answer

$P=(2\lambda+1,\ \lambda+3,\ 2\lambda+2)$, $Q=(\mu+2,\ 2\mu+2,\ 3\mu+3)$. Requiring $\vec{PQ}\parallel(1,-1,-2)$ gives $\lambda=\mu=3$:
$$P=(7,6,8),\qquad Q=(5,8,12).$$
$$PQ=\sqrt{4+4+16}=\sqrt{24}=2\sqrt6.$$

Solutions · 1