If the straight lines x=1+s, y=-3- s, z=1+ s and x= t2, y=1+t, z=2-t, with parameters s an…

Question

If the straight lines $x=1+s,\ y=-3-\lambda s,\ z=1+\lambda s$ and $x=\dfrac t2,\ y=1+t,\ z=2-t$, with parameters $s$ and $t$ respectively, are coplanar, then $\lambda$ equals

Accepted Answer

Line 1: point $(1,-3,1)$, direction $(1,-\lambda,\lambda)$. Line 2: point $(0,1,2)$, direction $(1,2,-2)$ (scaling $(1/2,1,-1)$ by 2). Difference of points: $(-1,4,1)$.
$$\begin{vmatrix}-1&4&1\1&-\lambda&\lambda\1&2&-2\end{vmatrix}=0\implies -1(2\lambda-2\lambda)-4(-2-\lambda)+(2+\lambda)=0\implies 5\lambda+10=0\implies\lambda=-2.$$

Solutions · 1