Question

Prove that the lines $\frac{x-1}{3}=\frac{y+1}{2}=\frac{z-1}{5}$ and $\frac{x+2}{4}=\frac{y-1}{3}=\frac{z+1}{-2}$ are skew.

Answer 1

$\frac{x-1}{3}=\frac{y+1}{2}=\frac{z-1}{5}$ and $\frac{x+2}{4}=\frac{y-1}{3}=\frac{z+1}{-2}$

If they are skew then there would not be any intersection point

Assume a point on line 1

$(3\lambda +1,2\lambda -1,5\lambda +1)$

Put in (2)

$\frac{3\lambda +3}{4}$ ___ (1) = $\frac{2\lambda -2}{3}$ _____ (2) = $\frac{5\lambda +2}{-2}$____ (3)

If by solving (1)&(2), (2)&(3) $\lambda$ value is different then lines do not intersect

(1) & (2) $\to 9\lambda +9=8\lambda -8$

$\lambda =-17$

(2) & (3) $\to -4\lambda +4=15\lambda +6$

$19\lambda =-2$

$\lambda =\frac{-19}{2}$

$-17\ne \frac{-19}{2}$

Therefore lines are skew