已知向量a=(cos3/2x,sin3/2x),向量b=(cos1/2x,-sin1/2x),x∈[-π/8,π/4].(1)求向量a·向量b(内积)及丨向量a+向量b丨;(2)若f(x)=向量a·向量b-丨向量a+向量b丨,求f(x)的最值.
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![已知向量a=(cos3/2x,sin3/2x),向量b=(cos1/2x,-sin1/2x),x∈[-π/8,π/4].(1)求向量a·向量b(内积)及丨向量a+向量b丨;(2)若f(x)=向量a·向量b-丨向量a+向量b丨,求f(x)的最值.](/uploads/image/z/8771702-14-2.jpg?t=%E5%B7%B2%E7%9F%A5%E5%90%91%E9%87%8Fa%3D%28cos3%2F2x%2Csin3%2F2x%29%2C%E5%90%91%E9%87%8Fb%3D%28cos1%2F2x%2C-sin1%2F2x%29%2Cx%E2%88%88%5B-%CF%80%2F8%2C%CF%80%2F4%5D.%EF%BC%881%EF%BC%89%E6%B1%82%E5%90%91%E9%87%8Fa%C2%B7%E5%90%91%E9%87%8Fb%EF%BC%88%E5%86%85%E7%A7%AF%EF%BC%89%E5%8F%8A%E4%B8%A8%E5%90%91%E9%87%8Fa%2B%E5%90%91%E9%87%8Fb%E4%B8%A8%EF%BC%9B%EF%BC%882%EF%BC%89%E8%8B%A5f%28x%29%3D%E5%90%91%E9%87%8Fa%C2%B7%E5%90%91%E9%87%8Fb-%E4%B8%A8%E5%90%91%E9%87%8Fa%2B%E5%90%91%E9%87%8Fb%E4%B8%A8%2C%E6%B1%82f%EF%BC%88x%EF%BC%89%E7%9A%84%E6%9C%80%E5%80%BC.)
已知向量a=(cos3/2x,sin3/2x),向量b=(cos1/2x,-sin1/2x),x∈[-π/8,π/4].(1)求向量a·向量b(内积)及丨向量a+向量b丨;(2)若f(x)=向量a·向量b-丨向量a+向量b丨,求f(x)的最值.
已知向量a=(cos3/2x,sin3/2x),向量b=(cos1/2x,-sin1/2x),x∈[-π/8,π/4].(1)求向量a·向量b(内积)及丨向量a+向量b丨;(2)若f(x)=向量a·向量b-丨向量a+向量b丨,求f(x)的最值.
已知向量a=(cos3/2x,sin3/2x),向量b=(cos1/2x,-sin1/2x),x∈[-π/8,π/4].(1)求向量a·向量b(内积)及丨向量a+向量b丨;(2)若f(x)=向量a·向量b-丨向量a+向量b丨,求f(x)的最值.
(1)
a.b
=(cos(3x/2),sin(3x/2)).(cos(x/2),-sin(x/2))
=cos(3x/2).cos(x/2)- sin(3x/2).sin(x/2)
=cos2x
(2)
|a+b|^2
=(cos(3x/2)+cos(x/2))^2+ (sin(3x/2)-sin(x/2))^2
= 2 + 2(cos(3x/2)cos(x/2)- sin(3x/2)sin(x/2) )
=2 + 2cos2x
= 2 + 2[2(cosx)^2 -1)]
|a+b| =2cosx
f(x) = a.b-|a+b|
= cos2x- 2cosx
= 2(cosx)^2-2cosx -1
= 2(cosx-1/2)^2 -3/2
min f(x) at cosx =1/2
min f(x) = -3/2
(1)
a.b
=(cos(3x/2),sin(3x/2)).(cos(x/2),-sin(x/2))
=cos(3x/2).cos(x/2)- sin(3x/2).sin(x/2)
=cos2x