设矢量 a = (x_1,...,x_n),b=(y_1,...,y_n)
则 = x_1 y_1 + ...+ x_n y_n
这样 ||^2 = ( ∑ x_i y_i )^2
由 Cauchy 不等式,( ∑ x_i y_i )^2 ≤ (∑ x_i^2) (∑ y_i^2)
而 |a|^2 = = ∑ x_i^2,|b|^2 = = ∑ y_i^2
故有 ||^2 ≤ ,或者等价地,|| ≤ |a| |b|.
而 = |a| |b| cos α,故有 |a| |b| |cos α| ≤ |a| |b|.
取 |a| |b| ≠ 0,就得到 |cos α| ≤ 1. 设矢量 a = (x_1,...,x_n),b=(y_1,...,y_n)
则 = x_1 y_1 + ...+ x_n y_n
这样 ||^2 = ( ∑ x_i y_i )^2
由 Cauchy 不等式,( ∑ x_i y_i )^2 ≤ (∑ x_i^2) (∑ y_i^2)
而 |a|^2 = = ∑ x_i^2,|b|^2 = = ∑ y_i^2
故有 ||^2 ≤ ,或者等价地,|| ≤ |a| |b|.
而 = |a| |b| cos α,故有 |a| |b| |cos α| ≤ |a| |b|.
取 |a| |b| ≠ 0,就得到 |cos α| ≤ 1.