find the equation for the plane that contains the line x=−1+3t , y=1+2t, z=2+4t and is perpendicular to the plane containing the two lines

l1: x=2t+1,y=-t-1,z=6t+5

l2: x=t+1,y=t-1,z=-3t+5

Let be the line given by the vector equation

.

First, we use the director vectors of the lines L1 and L2 to get the

vector equation of the plane containing them, which we denote by . This is,

We observe that . Therefore, the vector equation of defines a plane and is a normal vector to

Finally, the vector equation for the wanted plane, which we denote by , is

Thus, if , then and since is parallel to , then it is perpendicular to .