What is a mathematical explanation of the connection between: (1) projecting vector a onto vector b and multiplying the projected length of a with the length of vector b, and (2) the sum of the products of the equivalent components of the two vectors?

I realise there is a duality between a 2-dimensional vector and a 1x2 matrix, which can be used to explain the computation of the dot product. But I have not seen a satisfactory mathematical derivation, and was wondering whether there is another, simpler mathematical explanation.

I realise there is a duality between a 2-dimensional vector and a 1x2 matrix, which can be used to explain the computation of the dot product. But I have not seen a satisfactory mathematical derivation, and was wondering whether there is another, simpler mathematical explanation.