In which direction does the ecliptic plane make an angle of 60 degrees with galactic plane? In which direction does the ecliptic plane make an angle of 60 degrees with galactic plane? Is it toward the galactic center? (It can have any 360 degree orientation.)
The line defining where the ecliptic plane and the Galactic plane coincide lies (approximately) between the points defined by right ascensions  and declinations  of 6h +23 and 18h -23.

This can be seen below - the plot is in celestial coordinates with the ecliptic plane marked (thicker solid line) and the Galactic plane clearly visible in the background map of dust extinction.



The second of these coordinates is close to, but not coincident, with the Galactic centre (marked Sgr A*). So the relative tilt is sort of at right angles to the Galactic centre at present.

Of course this will change over tens of millions of years as the solar system orbits the Galaxy.

Edit: Another way of specifying the direction is just to ask where the ecliptic north pole is. It is at RA=18h, Dec=66d 34m, in the constellation of Draco.

A further way of thinking about it is to consider at which points in the sky do the ecliptic and Galactic planes have their maximum angular separation. These are 90 degrees away from the two points I defined above and are almost on the celestial equator at about 0h and 12h.