The standard definition for dot product of two vectors $\vec a=[a^i]$ and $\vec b=[b^i]$ (notice upper index) (e.g. in wiki) is right for contravariant vectors :
$$ \vec a \cdot \vec b = a^1 b^1 + \dots + a^n b^n $$
Questions: But is this definition is correct for covariant vectors $\vec a=[a_i]$ and $\vec a=[a_i]$ (notice bottom index) ? If no, then what is right definition ?
The answer is NO - this definition is not correct for covariant vectors - the right formula is
$$ \begin{array}{lll} \vec a\cdot \vec b &=& (g^{11}a_1+ g^{12}a_2+\dots+g^{1n}a_n)(g^{11}b_1+ g^{12}b_2+\dots+g^{1n}b_n)\\ &+& (g^{21}a_1+ g^{22}a_2+\dots+g^{2n}a_n)(g^{21}b_1+ g^{22}b_2+\dots+g^{2n}b_n) \\ &+& \dots\\ &+& (g^{n1}a_1+ g^{n2}a_2+\dots+g^{nn}a_n)(g^{n1}b_1+ g^{n2}b_2+\dots+g^{nn}b_n) \\ \end{array} $$
where $g^{ij}$ is contravariant form of metric tensor