a1=(a1⋅e1)e1a2=(a2⋅e1)e1+(a2⋅e2)e2a3=(a3⋅e1)e1+(a3⋅e2)e2+(a3⋅e3)e3⋮an=(an⋅e1)e1+(an⋅e2)e2+⋯+(an⋅en−1)en−1.(7.5)\begin{aligned} a_1 &= (a_1\cdot e_1) e_1\\ a_2 &= (a_2\cdot e_1) e_1 + (a_2\cdot e_2) e_2\\ a_3 &= (a_3\cdot e_1) e_1 + (a_3\cdot e_2)e_2 + (a_3\cdot e_3) e_3\\ &\vdots\\ a_n &= (a_n\cdot e_1) e_1 + (a_n\cdot e_2) e_2 + \cdots + (a_n\cdot e_{n-1}) e_{n-1}. \end{aligned}\tag{7.5}a1​a2​a3​an​​=(a1​⋅e1​)e1​=(a2​⋅e1​)e1​+(a2​⋅e2​)e2​=(a3​⋅e1​)e1​+(a3​⋅e2​)e2​+(a3​⋅e3​)e3​⋮=(an​⋅e1​)e1​+(an​⋅e2​)e2​+⋯+(an​⋅en−1​)en−1​.​