No. Unearth is an ability that you can activate while the creature card is in a graveyard, it is not cast from your hand and will not affect Vengevine.
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An ability exists on the stack independently of its source. Removal of the source after that time won't affect the ability. Some abilities cause a source to do something rather than the ability. Any activated or triggered ability that references information about the source checks that information when the ability is put onto the stack. Otherwise, it will check that information when it resolves. If the source is no longer in the zone it's expected to be in, its last known information is used.
Anytime you play a creature card, not when you use a special ability to bring that creature back or anything like that. If a card allows you to cast a spell without paying it's mana cost, it will say so. Play is the same thing as casting, but casting a creature such as nest invader only counts as 1 creature even though it puts a spawn token into play.
701.4a To cast a spell is to take it from the zone it’s in (usually the hand), put it on the stack, and pay its costs, so that it will eventually resolve and have its effect. A player may cast a spell if he or she has priority. See rule 601, “Casting Spells.”
Most of the time spells are cast from the had but certain keyword abilties like Rebound and flashback enable spells to be cast from odd places the exile and graveyard respectively.
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Every normal man must be tempted, at times, to spit on his hands, hoist the black flag and start slitting throats.
- H.L Mencken
I Became insane with long Intervals of horrible Sanity
All Religion, my friend is simply evolved out of fraud, fear, greed, imagination and poetry.
- Edgar Allan Poe
601.2. To cast a spell is to take it from where it is (usually the hand), put it on the stack, and pay its costs, so that it will eventually resolve and have its effect...
Some spells may be cast from outside the hand due to other effects in play. For example, if you attack and connect with Thada Adel, Acquisitor, you could choose an Ornithopter and cast it, and that would count towards bringing back vengevine. Likewise, cascading into a creature and casting it via cascade would also count.
The previous definitions are quite vague: "cast" a spell means:
- put the spell on the stack (you declare to play the card)
- choose the modes for the spell (like Esper Charm)
- choose any target for the spell
- distribute something to the recipients (damage from Ignite Disorder)
- choose to pay any additional costs (e.g. Kicker) and, if any, to use alternative costs (Allosaurus Rider)
- calculate the total cost
- pay the costs, in any order (most of the times, you pay just mana; compare Harrow)
Although most of the times you'll play (=cast) cards from your hands, some abilities let you cast spells from other zones: for example, Flashback or Suspend. This abilities read "cast" in their oracle or glossary text (pre-M10 cards have "play" printed instead of "cast", but that's exactly the same). In this cases, the card always passes on the stack.
Unhearth doesn't say, as pointed out before, "cast": it's an ability which can be activated only from graveyards and allows you to put the creature directly on the battlefield. When an ability reads "put ... onto the battlefield", that doesn't count for casting spells.