Vector algebra

1. $\underline{\boldsymbol{ a}}=a_i \underline{\boldsymbol{ e}}_i$ and $\underline{\boldsymbol{ b}}=b_i\underline{\boldsymbol{ e}}_i$. Their scalar product is $\underline{\boldsymbol{ a}}\cdot\underline{\boldsymbol{ b}}=a_i b_j \underline{\boldsymbol{ e}}_i \cdot \underline{\boldsymbol{ e}}_j$, where $\underline{\boldsymbol{ e}}_i \cdot \underline{\boldsymbol{ e}}_j=\delta_{ij}$ following the orthonormal coordinate system. Hence, $\underline{\boldsymbol{ a}}\cdot\underline{\boldsymbol{ b}}=a_i b_i$ (Normally, we use this without any derivation.)
2. We require that $\underline{\boldsymbol{ e}}_i\cdot\underline{\boldsymbol{ e}}_j=\delta_{ij}$, that is that $|\underline{\boldsymbol{ e}}_i|=1$ and $\underline{\boldsymbol{ e}}_i$ is perpendicular to $\underline{\boldsymbol{ e}}_j$ if $j\neq i$. Furthermore, we require that it is right-handed, i.e. $\underline{\boldsymbol{ e}}_3=\underline{\boldsymbol{ e}}_1\times\underline{\boldsymbol{ e}}_2$.