# Thread: Finding the equation of a plane that contains two lines

1. ## Finding the equation of a plane that contains two lines

I need help finding the equation a plane that contains the lines <t,2t,3t> and <3t,t,8t>
I don't really know where to start. I tried setting the two lines equal to each other to find a point of intersection but it turned out funky. Any help would be appreciated.

2. ## Re: Finding the equation of a plane that contains two lines

I found the solution in case anyone else has the same problem.

Lines intersect at origin O(0,0,0)
r: [t,2t,3t] = t[1,2,3] = t u
s: [3t,t,8t]=t[3,1,8] = t v

where
u = [1,2,3]
v = [3,1,8]

are the direction vectors of the lines

a vector which is normal to the plane containing r and s is
the cross product uxv

u x v = det([i, j, k; 1, 2, 3; 3, 1, 8])

u x v = 13i + j - 5k = [13,1,-5]

so the plane containing r and s has equation

13x + y - 5z = 0

3. ## Re: Finding the equation of a plane that contains two lines

Originally Posted by jtl4231
I need help finding the equation a plane that contains the lines <t,2t,3t> and <3t,t,8t>
I don't really know where to start. I tried setting the two lines equal to each other to find a point of intersection but it turned out funky. Any help would be appreciated.
It is clear that if $t=0$ then $(0,0,0)$ is on both lines.

Now the vector $<1,2,3>\times<3,1,8>=<13,1,-5>$ is the normal to the plane.