# 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?

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.
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eishale2n
$\stackrel{\to }{a}\cdot \stackrel{\to }{b}=ab\mathrm{cos}\theta$
Also, suppose that we have an orthonormal basis $\left\{{\stackrel{^}{e}}_{i}\right\}$. Then
$\stackrel{\to }{a}=\sum _{i}{a}_{i}{\stackrel{^}{e}}_{i}\phantom{\rule{0ex}{0ex}}\stackrel{\to }{b}=\sum _{i}{b}_{i}{\stackrel{^}{e}}_{i}$
Now using the geometrical definition, if two of the basis vectors are the same
${\stackrel{^}{e}}_{i}\cdot {\stackrel{^}{e}}_{i}={e}_{i}{e}_{i}\mathrm{cos}0=1$
and if two vectors are different
${\stackrel{^}{e}}_{i}\cdot {\stackrel{^}{e}}_{j}={e}_{i}{e}_{j}\mathrm{cos}\frac{\pi }{2}=0$
Then
$\stackrel{\to }{a}\cdot \stackrel{\to }{b}=\stackrel{\to }{a}\cdot \left(\sum _{i}{b}_{i}{\stackrel{^}{e}}_{i}\right)=\sum _{i}\left(\stackrel{\to }{a}\cdot {\stackrel{^}{e}}_{i}\right){b}_{i}=\sum _{i}{a}_{i}{b}_{i}$