That vector is what must be conserved.

Putting it this way blew my mind a little bit, for some reason. Great answer.

I’m sorry but this answer is not wrong, it’s just not an answer.

You are not explaining the reason why, you are explaining how to solve it using some useful tricks.

Physics doesn’t act according to your thumb and surely your thumb does not encapsulate the essence of inertia.

It’s akin to answering “What is 2 x 3?” and answering providing a table where to look up the answer for each combination of number to multiply.