Google suggests that the A380 engines produce a total thrust of 130,000 kg roughly. If the thrust angle was 20 degrees the vertical component would be
130,000 * sin(20) = 44,000Kg
Not even that much, since most planes don't rotate much beyond 10 degrees, and certainly not airliners unless they have cast-iron tails (to handle the ensuing tailstrike). Go ahead and try rotating directly to a 20-degree pitchup on the AH the next time you take off and see what happens....
The typical takeoff and climb angles of all Boeing planes - Bangalore Aviation
More like 10 degrees, which gives 22,100 lbs of "vertical" thrust vector in that calculation.
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To me, it just isn't that mysterious or complicated:
First - wings don't just suddenly begin producing lift at V1 or Vr - they are producing
some lift at any forward airspeed. The faster you go, the more lift they produce. At just below Vr, they may be producing lift equal to, say, 80% of the aircraft weight, with the remaining 20% keeping the wheels on the ground.
At take-off, you rotate the plane, which increases AoA. Increasing AoA from 0 to 10 degrees increases a generic wing's lift coefficient nearly 3x:
File:Lift curve.svg - Wikipedia, the free encyclopedia
3x increase in lift means lift increases quickly from 80% to 240% of aircraft weight - and up you go.
Even a 2° increase in AoA increases lift nearly 50% (and the Buff in that video does rotate about 2° - the nose wheel comes up at about 0:15) - which is enough to increase lift from our assumed 80% of weight, to 120% of weight.
or put even more simply:
Condition 1 - airplane weight > than lift = airplane stays on ground
Condition 2 - airplane weight < than lift = airplane climbs
A plane takes off when the increased AoA of rotation, and subsequent increase in lift, moves the plane from condition 1 to condition 2.