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"Descending on a given glide slope (e.g. ILS) at a given airspeed— is the size of the lift vector different in headwind versus tailwind?" -- the variation in the size of the Lift vector as the still-air glide ratio changes comes into play here, exactly as it does in the present answer

"What produces Thrust along the line of flight in a glider?"

"Does lift equal weight in a climb?"

Descending on a given glide slope (e.g. ILS) at a given airspeed— is the size of the lift vector different in headwind versus tailwind? -- the variation in the size of the Lift vector as the still-air glide ratio changes comes into play here, exactly as it does in the present answer

What produces Thrust along the line of flight in a glider?

Does lift equal weight in a climb?

No, the power requirements are generally not exactly identical in the powered case and the gliding case, because the Lift vector must be slightly larger in the powered case, which implies that the angle-of-attack must be slightly higher in the powered case. Therefore the Drag vector is unlikely to be identical in size between the two cases. But for most practical purposes, in aircraft with reasonably high L/D ratios, the difference in Lift, Drag and power required between the gliding and powered cases is negligible.

No, the power requirements are generally not exactly identical in the powered case and the gliding case, because the Lift vector must be slightly larger in the powered case, which implies that the angle-of-attack must be slightly higher in the powered case. Therefore the Drag vector is unlikely to be identical in size between the two cases. But for most practical purposes, in aircraft with reasonably high L/D ratios, the difference in Lift, Drag, and power required between the gliding and powered cases is negligible.

The original question was phrased to hold airspeed constant between the gliding and powered cases, not angle-of-attack. To hold the airspeed constant as we transition from the gliding case to the powered case, we must increase the angle-of-attack to increase the lift coefficient to provide the extra lift needed in the powered case. Now we can be sure that the L/D ratio is not remaining constant. Since airspeed is now constant, the change in the required power will be directly proportional to the resulting change in the Drag vector. And the resulting change in the Drag vector depends upon where we are on the L/D versus airspeed curve. If we are in high-speed cruise well above the best L/D speed, the Drag vector will actually be smaller in level flight than in gliding flight. If we are flying slower than best L/D speed, the Drag vector will be larger in level flight than in gliding flight. There is a special case where we happen to gliding with an angle-of-attack slightly lower than the angle-of-attack yielding the maximum L/D ratio and minimum Drag, and as we transition to horizontal flight, we increase the angle-of-attack and end up with an angle-of-attack slightly higher than the angle-of-attack yielding the maximum L/D ratio and minimum Drag, and the net change of Drag is exactly zero.

The original question was phrased to hold airspeed constant between the gliding and powered cases, not angle-of-attack. To hold the airspeed constant as we transition from the gliding case to the powered case, we must increase the angle-of-attack to increase the lift coefficient to provide the extra lift needed in the powered case. Now the L/D ratio is almost certainly not remaining constant. Since airspeed is now constant, the change in the required power will be directly proportional to the resulting change in the Drag vector. And the resulting change in the Drag vector depends upon where we are on the L/D versus airspeed curve. If we are in high-speed cruise well above the best L/D speed, the Drag vector will actually be smaller in level flight than in gliding flight. If we are flying slower than best L/D speed, the Drag vector will be larger in level flight than in gliding flight. There is one special case where we happen to gliding with an angle-of-attack slightly lower than the angle-of-attack yielding the maximum L/D ratio and minimum Drag, and as we transition to horizontal flight, we increase the angle-of-attack and end up with an angle-of-attack slightly higher than the angle-of-attack yielding the maximum L/D ratio and minimum Drag, and the net change of Drag is exactly zero.