It should be long enough to extend below the swingarm and engine. I cut the ends if they have slits. The acid fumes will eat the paint and metal.

The MotoBatt MBT6N4 is a sealed absorbed glass mat type. No fumes. It fits the DT3.

There may be a metal loop welded to the middle front of the swingarm. If so run the battery vent line through there. It keeps the tube centered and anything coming out should hit the tire and not zinc or paint.

I assume it should be like my DT2 battery vent tube is routed.
Drops out of the battery and behind the 2 Stroke Oil Tank. The hose goes through the hole in the black bracket (see pic) under the oil tank from the rear to the front, then down to in front of the swing arm.






I've more (better) pics on my home PC, but I'm not home this week. Let me know if you'd like to see more and I'll put them up when I'm back.

Almost 100% positive the battery vent tube is going through the bracket. I'll check for sure once home. The original owner hadn't replace the OEM battery as far as I could tell.

No on the oil vent tube. It goes into the small hole on top of the clutch actuator cover. That puts it directly over the chain so any spillage goes onto the chain. One of Yamahas’ better ideas.

As others have said. Battery vent tube down thru bracket hole under oil tank then in front of swingarm. Oil tank vent tube down the right side of oil tank then into small hole in left side engine cover by chain. 

Here's a picture that might help. It's a 73 RT3 but the same as your bike. You can clearly see the battery vent tube. 






 

I took pictures of about everything as I tore the bike down last year. I found this one in my phone:

You can see the battery vent tube still attached in this pic coming through the bracket.