Question

Is the direction of acceleration Is same as the direction of
velocity​

Answer 1

No, it does not.

First of all, there is a mistake in your formula. Correct one is a⃗ =dv⃗ dt. (Note, there is no m in here). And this is just a definition of acceleration. This formula is NOT derived from F⃗ =ma⃗ . As I said it's just a definition of acceleration.

There is no reason why dv⃗ dt and v⃗ should have the same direction. We have a body, it's velocity is v⃗ at the moment. A little bit later it's velocity is a little different: v⃗ +v⃗ 1. The difference v⃗ 1 can be directed anywhere, isn't it? For simplicity let's assume the acceleration is constant. In this case the difference is proportional to time: a⃗ ∗dt. So, the direction of a⃗ is the direction of difference of velocity. It can have any direction!