Nib asked, and Nib — askNib's tutor — drew the answer live at a whiteboard. This is the spoken transcript; enable JavaScript to watch it drawn.
why do planets orbit in a flat plane
Planets circle the Sun in almost one flat sheet. Yet gravity alone does not demand that: it would happily allow orbits tilted at any angle at all.
And the real planets? Measured against Earth's orbit, none tilts more than seven degrees, and most less than three. That is a flat sheet, and it needs a reason.
Go back 4.6 billion years. The Sun's birthplace was a huge, slow cloud of gas and dust, with its clumps drifting in every direction at random.
Gravity pulls the cloud in, and it spins faster, like a skater pulling in her arms. Clumps collide: their up-down motion cancels, but the shared spin stays.
What is left is a thin spinning pancake with the young Sun in the middle. Planets grew out of that disk, so they all circle in its plane, in its direction.
So the flat plane is a fossil. Collisions killed the up-down motion, the spin survived, and planets formed in the disk. Comets from the far edge, though, still come from every angle.
The planets' flat sheet is a fossil: the shape of the cloud that spun them into being.