- 求向量a*b及向量|a+b|;若fx=向量a*b
- 已知向量a=(c3x/2,sin3x/2),b=(cosx/2,-sinx/2),且x属于[0,t/2]
- 已知向量a=(c3x/2,sin3x/2),b=(cosx/2,-sinx/2),且x∈[0,π/2], 求a•b及|a+b|；若f(x)=a•b-2r|a+b|的最小值是-3/2,求r的值
a•b = (cos3x/2,sin3x/2)•(cosx/2,-sinx/2)
= cos(3x/2)cos(x/2) - sin(3x/2)sin(x/2)
= cos(3x/2+x/2) = cos(2x)
|a+b| = √[(cos3x/2+cosx/2)²+(sin3x/2-sinx/2)²]
= √[(2cosxcos(x/2))²+(2cosxsin(x/2))²]
= 2|cosx|√[cos²(x/2)+sin²(x/2)]
= 2cosx ......... x∈[0,π/2]
f(x) = a•b-2r|a+b|
= cos(2x)-2rcosx
= 2cos²x-2rcosx-1
= 2(cosx-r/2)²-(1+r²/2) 的最小值是-3/2
r≤0时，f(x)的最小值=f(π/2)=-1 ≠-3/2,舍去
r≥2时，f(x)的最小值=f(0)=1-2r≤-3，舍去
0≤r≤2时，f(x)最小值 = -(1+r²/2)=-3/2--->r=1