例3应用向量证明 Cauchy--Schwarz不等式 a+a2b2+a3b3a2+a2+a2·++b 证记a={a1,a2,a3}b={1,b2,b3} 则|a=√a2+n2+n2|b}=√研2+b+b d·b=a1b1+a2b2+a3b3 →|ab同=a|·b|·|cos(a,b) ≤a||b|=√a2+a2+a +b2+b →|a1b1+a2+ah2a2+a2+n2所2+b2+b例3 应用向量证明Cauchy—Schwarz不等式 2 3 2 2 2 1 2 3 2 2 2 1 1 2 2 3 3 1 | a b + a b + a b | a + a + a  b + b + b 证 记 a = a1 ,a2 ,a3  b = b1 ,b2 ,b3  则 2 3 2 2 2 1 | a |= a + a + a  2 3 2 2 2 1 | b |= b + b + b  a b = a1b1 + a2b2 + a3b3    | a b | | a | | b | | cos(a,b)|         =   | a | | b |     2 3 2 2 2 1 2 3 2 2 2 1 = a + a + a  b + b + b 2 3 2 2 2 1 2 3 2 2 2 1 1 2 2 3 3 1  | a b + a b + a b | a + a + a  b + b + b