I'm guessing you mean you want to plant 3 rows of tomato plants one in front of another? That could create shading issues no matter how far apart they are. Will they be running N-S or E-W? That will determine at least some of the exposure to sun. So will the angle of the sun relative to your location and any surrounding structures.

Row spacing has to allow not only for room for the plants but for access for you to care and harvest them so I'd estimate you'd need at least 6 feet apart and 8 feet would be better.

If you can stagger them rather than putting one right in front of another it would be better.

Dave

First : It depend largely on the orientation. If, for example, the rows are N<>S, then the spacing can be anything, just to allow you to walk between the rows.

But if the rows run E<>W then you'll need more spacing/walkway width. But it also depends how many hours of direct sun you get. Solar Altitude get quite high from May till end of August. That means the plants will cast short shadow during the mid day hours. My own raised beds run both N,<>S, and E<>W and I only get 5 hours of direct sun. Not and ideal situation , but it works ok.

Sey