Question

geometric interpretation of multplying vectors?

Answer 1

There is an arrow above P_0P. It means "vector that send P_0 to P".

Technically, P-P_0 don't make sense because they are points, not vectors, so you can't subtract them. But we understood it to means the subtraction between their position vector, and position vector is defined once you choose an origin. But writing P_0P with an arrow on top is a more neutral notation that does not assume you picked an origin.