If we use r1,r2,r3 then how would the vector operator operator operate on it?Like it didnt in A when we used A1,A2,A3.It's a matter of notation: we're just giving names to the three components of both vectors.
You can replace them by ##r_1, r_2, r_3##, if that makes you feel any better. In general, if v is a vector, it is customary to denote its components by v1, v2, v3. However, if r is the position vector, then (x, y, z) is also quite common.
Also note that though the fact that one is named r hints that it comes from a physical application in which a position vector is involved, the mathematical identity actually holds for any two vectors u, v.
I'm sorry - the example in your attachment very clearly states thatNo its not..here at least