Example - skew lines

Example — Skew lines

Let $r_1: (x,y,z)=(-1,-1,5)+t(2,-1,0)$ and $r_2: (x,y,z)=(5,5,-3)+s(2,-1,-1)$.

Parallelism. The direction vectors $(2;-1;0)$ and $(2;-1;-1)$ have third component $0\ne -1$: they are not parallel.

System. $\begin{cases}-1+2t=5+2s\\-1-t=5-s\\5=-3-s\end{cases}$ From the third: $s=-8$. From the second: $t=-1-5+(-8)=-14$. Check on the first: $-1+2(-14) = -29$ whereas $5+2(-8)=-11$, and $-29\ne -11$: inconsistent ⚠.

Conclusion. The lines are skew.

Links

Topics: Analytic geometry in space
Concepts: Line · Line in space · Skew lines
Skills: Analytic geometry · Reasoning by cases · Solving systems