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When a airliner takes off, it looks like after rotation and lift-off, the plane flies up in the air parallel to the length of its fuselage.

Assuming lift from the wings is the major force in taking off, that should not necessarily be the case. (E.g. during landing the fuselage is typical at an angle with the movement of the aircraft.)

So, I am wondering, is my observation wrong (i.e. do airplanes not take off parallel to their major axis)?

Or is the engine thrust a greater factor in taking off, since the thrust vector is by design always parallel to the fuselage.

Or, 3rd option, is the airplane designed such that the lift of the wings in the default take off configuration results in a parallel flight, e.g. because this is most efficient from aerodynamic POV?

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The Angle of Attack, which is roughly the delta between the fuselage and the velocity vector, is on the order of 5-10 degrees shortly after takeoff. This difference from exact alignment of the vector and the fuselage would be hard to observe from any single fixed point, with the aircraft moving past the observer pretty rapidly, but it is real - and very important to the flight of the aircraft!

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Engine thrust a greater factor in take off ... most efficient from aerodynamic point of view

Aircraft need to overcome drag with thrust in level flight. When an aircraft climbs, it needs more thrust.

The easiest way to understand this is to point the aircraft/rocket in a 90 degree (vertical) climb. The wing no longer helps at all, and thrust alone must match gravity plus drag from forward motion.

A winged aircraft in level flight needs much less thrust because the wing is far more efficient at creating lift to counteract gravity.

As the aircraft raises the nose to climb, the lift vector tilts away from vertical. Vertical lift is lost by cosine degrees climb. A 10 degree climb loses only 1.5 % of the vertical lift but requires 17 % (sine 10 degrees) more thrust to compensate in addition to thrust required to maintain airspeed.

One can see thrust requirement will increase as angle of climb increases.

In order to be most efficient from the aerodynamic point of view, the aircraft raises its nose a bit further than the line of flight to maintain its best angle of attack. This is done in level flight, climbing, or descending.

the difference is the amount of thrust required to maintain that straight line of flight and AoA in all cases

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Or, 3rd option, is the airplane designed such that the lift of the wings in the default take off configuration results in a parallel flight, e.g. because this is most efficient from aerodynamic POV?

This is your best guess so far, but not exactly how I would describe the attitude.

Landing Configuration and High AOA

Remember, airplanes almost always use a different wing configuration for landing. Along with the extra lift and drag provided by flaps, there is a change in angle of attack and a corresponding change in the required pitch angle. Because the angle of attack needs to be on the high side to maintain a low speed, the fuselage doesn't point downward unless full flaps are used. On some airplanes the full flap range is not needed or even detrimental to manuevering, so in a sense there's an advantage to flying more nose up rather than "parellel" to the flight path.

Relative Wind During Climbout and Low AOA

Relative wind is tilted down (from the upward direction) during a climb, so even with a low angle of attack, the pitch angle is above the horizon and still close to the flight path angle. For turbojet aircraft, the climbout profile involves increasing airspeed and making configuration changes that influence the needed pitch angle. You might observe slight pitch changes during the initial climb because the acceleration and changes of configuration are changing the angle of attack and changing the rate of climb. Those happen even in stable weather.

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True, but when you are landing , you decrease gradually the speed on final, that alone will increase the AOA, then aerobrake,flying parallel to the runaway, visually the nose is upward, but when you take off the speed "is alive", increasing, the aircraft become" clean "landing gear up, flaps up, no needed for much of Angle of Attack. Also the parallax have a great role in view on this instance.

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