show that the lines are all in a plane and find the equation of the plane.

Question

Three lines $L_1,L_2,L_3$ pass through the origin with the parallel vectors $$V_1=i+2j-k$$$$V_2=3i+5j+7k$$$$V_3=2i+3j+8k$$ are given,show that the lines are all in a plane and find the equation of the plane.

The parametric equation of the lines :

$$x_1=t,y_1=2t,z_1=-t$$ $$x_2=3t,y_2=5t,z_2=7t$$ $$x_3=2t,y_3=3t,z_3=8t$$

But I don't know how to continue.

Answer 1

Note that $$V_1+V_3=V_2$$ Since one of the vectors is a linear combination of the others, therefore, all of these are coplanar.

Answer 2

hint

$$V_1 + V_3 = V_2$$

and

$(V_1,V_3) $ are independant, thus The vectorial plane defined by $ (V_1,V_3) $ contains also the line given by $ V_2$.

Let us look for the normal vector $N= (a,b,c) $ to the plane.

So $$N•V_1=a+2b-c=0$$ $$N•V_3=2a+3b+8c=0$$ The equation of the plane will be

$$ax+by+cz=0$$ with $$b-10c=0$$ $$a=c-2b=-19c$$

So, if we take $c=1$, we find $$-19x+10y+z=0$$