Why do we use different H values in a vector Pedersen Commitment?

Question

The usual Pedersen Commitment looks like such:

C = xG + aH

Where x and a are scalars.

However, we can also commit vectors, s.t.

C = xG + (a_1 * H_1 + a_2 * H_2)

Could anyone point out why we change the H values for each vector value (a_1 and a_2)? To my knowledge, having the same H for a_1 and a_2 would still be perfectly hiding.

a_1 means 'a subscript 1' denoting the first scalar value in the vector a.

I believe H_i is calculated s.t. H_i = Hash(G + i) * G

Answer 1

It is correct that using the same H each time would be perfectly hiding. However, the job of a Pedersen Commitment is not only to hide, but to commit.

If you used the same H each time, then all you would be committing to would be the sum of all components. You could later pretend that you had committed to a different number of components or to different component values as long as they still added up to the same amount.