Question

Show that the two lines $\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}$ and $\frac{x-4}{5}=\frac{y-1}{2}=z$ intersect. Also find the point of intersection of these lines.

Answer 1

In line 1 : $\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}$

Let $\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}=\alpha$

$x=2\alpha +1,y=3\alpha +2,z=4\alpha +3-\left(1\right)$

In line 2 : $\frac{x-4}{5}=\frac{y-1}{2}=\frac{z}{1}$

Let $\frac{x-4}{5}=\frac{y-1}{2}=\frac{z}{1}=\alpha$

$x=5\alpha +4,y=2\alpha +1,z=\alpha -\left(2\right)$

Comparing (1) & (2),

$\Rightarrow -3\alpha =3$

$\alpha =-1$ (from comparing x co-ordinates)

$2\alpha +1=3\alpha +2$

$\alpha =-1$ (comparing y co-ordinates)

$4\alpha +3=\alpha$

$\alpha =-1$ ( comparing z co-ordinates)

$\therefore$ they intersect

putting $\alpha$ in (1)

$x=-2+1=-1$

$y=3(-1)+2$

$=-1$

$z=4(-1)+3$

$=-1$

$(x,y,z)=(-1,-1,-1)$ is the point of intersection