1.$sumlimits_{i=1}^{n}sumlimits_{i=1}^{n}{a_ib_j}=sumlimits_{i=1}^{n}sumlimits_{i=1}^{n}{a_jb_i}=
left(sumlimits_{i=1}^{n}a_i
ight)left(sumlimits_{i=1}^{n}b_i
ight)$
2.$sumlimits_{i=1}^{n}a^2_i+2sumlimits_{1le i<jle n}a_ia_j=left(sumlimits_{i=1}^{n}a_i
ight)^2$
3.$nsumlimits_{i=1}^{n}a^2_i-left(sumlimits_{i=1}^{n}a_i
ight)^2=sumlimits_{1le i<jle n}(a_i-a_j)^2$
4.(拉格朗日恒等式)
$left(sumlimits_{i=1}^{n}a^2_i
ight)left(sumlimits_{i=1}^{n}{b^2_i}
ight)-left(sumlimits_{i=1}^{n}{a_ib_i}
ight)^2
=sumlimits_{1le i<jle n}(a_ib_j-a_jb_i)^2$