Von Klinkerhoffen;
Sorry have to disagree , I think you are confusing a stalled wing due to excessive AoA with one that is not producing lift due to insufficient airspeed
Yes. My bad. There will be no wing stall. Stall comes as a result of too-high AoA, not insufficient velocity. In straight-and-level flight, the two are linked, as minimum amount of lift is required to balance weight. However, in the theorized straight vertical ascent, thrust is balancing weight, so wing-lift is what it is. When one examines non-traditional scenarios, one must also re-examine assumptions taken for granted.
ft;
In a steady state vertical climb, the lift will most definitely be zero. As you say, the lift is perpendicular to the trajectory through the air. With no opposing force, the trajectory would not remain vertical very long if the wing indeed generated lift. With lift, a vertical climb is not steady state.
You will have to reduce AoA to the zero-lift AoA for the climb to remain vertical. At the zero-lift AoA, the lift coefficient is zero and... and here it comes... there's zero lift.
Sorry. I have to disagree. It is true that the trajectory would not remain vertical very long if the wing generated lift. And it is true that you will have to reduce AoA to the zero-lift AoA in order to achieve this. However, unless you have an all-moving wing, how do you change your AoA without adjusting your climb angle? Remember the constraint on this theoretical problem: straight vertical climb. If you adjust your climb angle to other than 90 degrees, you then have the probem of a thrust component not in the vertical direction, which has to be balanced by a force perpendicular to your trajectory.
All I'm trying to say here is that for most applications, this straight vertical climb we're talking about does produce some wing lift. You could have zero-lift in the case where the wing zero-lift angle is parallel to the aircraft longitudinal axis, but very few aircraft are designed this way.