Here is a "manual" method:
select the line and sbdivide it in edit mode, select the newly created vertex and press Ctrl G to assign it to a new vertex group.
Add a shrinkwrap modifier, limit its action to the new vertex group, change its wrap method to "Project" and snap mode to "On surface".
Enable "Negative" and "Positive" and pick your solid mesh as Target.
In object mode Apply the modifier.
Now you can select the vertex in the line and read its coordinates in the Transform tab of the "N" properties panel.