Find an equation for the plane that contains the two (parallel) lines l(t) = (0, 1, −2) + t(2, 3, −1) and s(t) = (2, −1, 0) + t(2, 3, −1).

Question

I'm don't know where to start on this question and need some assistance.

Answer 1

Let $A(0,1,-2)$, $B(2,-1,0)$ and $\vec{n}(a,b,c)$ be a vector normal of the plane.

Thus, $\vec{n}\perp\vec{AB}$ and $\vec{n}\perp(2,3,-1)$.

Now, $\vec{AB}(2,-2,2)$ and we got the following system. $$a-b+c=0,$$ $$2a+3b-c=0,$$ which after summing gives $3a+2b=0$ and we see that $a=2$, $b=-3$ is valid.

Hence, we can assume that $\vec{n}(2,-3,-5)$ and we get an equation of the plane: $$2(x-0)-3(y-1)-5(z+2)=0$$ or $$2x-3y-5z-7=0.$$