What are the challenges of finding a plane in 4-space using two given lines?

[ctrlaltdel121]

Homework Statement

I am given two lines in vector form in 4-space. I need to write an equation for the plane that is parallel to one line, and contains the other line.

Homework Equations

well i know that in 3d I would find a normal vector for the plane that would be perpendicular to both lines, and that would let me define the plane. However i am stumped because in 4-space there are an infinite amount of vectors that are perpendicular to both lines.

The Attempt at a Solution

I took the dot product of the vectors and got an infinite number of possible normals. I know one point on the plane from the line that is contained in the plane, all i need is another vector in the plane to define it but I cannot figure it out.

I didn't want to give the numbers here because i can find it out on my own once I am given some direction.

 

[CompuChip]

However i am stumped because in 4-space there are an infinite amount of vectors that are perpendicular to both lines.

The same is true in three dimensions: all the vectors (0, 0, z) for z not equal to 0 are perpendicular to the (x, y) plane.
In this case, the space of vectors which is perpendicular to both lines is just two-dimensional, instead of one. For example, for the (x, y) plane the perpendicular vectors would be (0, 0, z, u) for z.u not equal to 0.