Thanks! Yes, when you grow in a direction at right angles to yourself, you don't increase in magnitude. So the turning arrow is the same length at the end as it was at the beginning.

This is actually a slightly tricky point, because it requires the direction of growth to change as you grow. If you grow by a finite amount all at once, then you do get slightly longer. So you might expect an outward spiral.

But a point moving round in a circle is always moving at right angles to the radius connecting it to the centre.

And in the same way, an arrow whose tip is moving at right angles to its shaft isn't extending. If the length is changing, then the direction isn't 90 degrees.

To reason about this properly I think you need some sort of theory of infinitesimals and continuous motion.