What is the vector equation for the line of intersection of two planes with the only given information being their distances from the origin?

Question

The question I'm working on is:

Two planes have non-parallel unit normals n and m and their closest distances from the origin are 3 and 7 respectively. Find the vector equation of their line of intersection.

I have calculated the implicit vector given by the cross product of the two normals, and from there derived a point on the line, but this is all done implicitly and seems rather vague. #

I'm thinking there may be something in the fact that the normals given are unit vectors, but I've become rather stuck.

Any help will be greatly appreciated.

Thanks

Answer 1

Hint: if $\vec r$ is a positional vector, then the equations of the planes are $\vec n \cdot \vec r=3$ and $\vec m \cdot \vec r=7$. Does this help?