Today when I ran the vette,i got white smoke coming out from under the hood.I opened the hood and the smoke was coming from the passenger side up threw the wiper tray drain. After the car warms up ,and I start driving it stops.Is this a sign of the heater core going up?

Can you smell antifreeze in the corkpit when you have the defroster on?  That is a dead giveaway. 

Only saw white smoke once and it was on my '68 Impala.  That was a teenagers failure to get the electricians tape all the way down the connection.  I smoked the dash wiring.  Hard to drive with a car full of smoke!

Check the passenger carpet. If the heater core let go, you'll usually get anti-freeze running down to the floor pan  - hence the smell....

As has been said, check the pass side floor for water. If it's wet there, then heater core has sprung a leak. You will usually get steam from the defrost vent, too, when the core leaks.
However....IF the floor is dry, and no steam comes from the vents, you may have an electrical issue that is letting the smoke out of the wires somewhere...heater cores will normally leak all the time...not just when cold. If the core is leaking slightly, it may be that the exhaust is burning off the coolant that drips on it overnight from the core leak, which could explain only seeing the steam/smoke when it's cold/first started.
Either way, it ain't good...you got some work to do!

No it's not electrical,Steam comes out of the deforster vent and the pass floor is wet I thought it was my weather striping going bad. Now I know.One more question if I valve off the heater lines that should keep it from leaking.I am replaceing the dash in the spring so I will also be doing a heater core too.

Yeah, you can remove the hoses and tie them together for the time being. I wouldn't recommend blocking the flow with a shut-off valve, even tho a lot of folks do that. I just feel it's better to let the coolant flow thru the hoses and water pump the way it was designed. Blocking the flow can effect the way the coolant flows thru the heads/block. That's just me...

If the hoses are old, you could pull them off, change the nipple in the intake so it is the same as the water pump nipple and use a new section of hose 5/8 or 3/4, whichever way you want to go so you keep the flow going (important in older men and older cars).

If the hoses are new, you could either pull them off and do as above, or get a union that is 5/8 on one end and 3/4 on the other.  That and a couple of clamps and you're good to go.

The location of the hoses on the heater core isn't the easiest to gain access to, especially if you don't have a lift or ramps to get her in the air so you can lay under her just behind the passenger side front wheel.  Or you can pull the wheel for easier access.

Good Luck!

How many hoses are going to the heater core? I have 2 that run into the fire wall and i can get to them without raising the car.Am I missing something?

Nope...just the two...one in, one out. If your car has A/C, the hoses run to the bottom of the A/C plenum. If you do not have A/C, then they'll run directly into the firewall area.

 Thanks that what i thought,but wanted to be sure.