Consider the xyz plane.

The x-axis and y-axis intersect.

And z-axis is perpendicular to both of them.

That is how the visuallization looks.

Line 1) Is algined with the vector,

Line 2) Is algined with vector,

The line that we seek is perpendicular to both, that is a vector that is orthogonal to both these non-zero vectors.

We can use the cross product,

.

We can remove the 7 multiple, because if it is algined with it is certainly aligned with .

Thus, the parametric equations are,