a=(cos3/2x,sin3/2x) b=(cosx/2,-sinx/2) c=(1,-1),x
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向量 a.向量b=cos3x/2*cosx/2-sin3x/2*sinx/2.
a.b=cos(3x/2+x/2).
=cos2x.
(1) 当a.b=1/2时,则cos2x=1/2.
2x=2kπ+π/3.
x={x| kπ+π/6,k∈R}.
(2) 设f(x)=(a-c)^2,.
向量a-向量c=cos3x/2-√3,sin3x/2+1).
(a-c)^2=(cos3x/2-√3)^2+(sin3x/2+1)^2.
=cos^2(3x/2)-2√3cos(3x/2)+3+sin^2(3x/2)+2sin3x/2+1.
=2(sin3x/2-√3cos3x/2)+5.
=2*2[sin3x/2*(1/2)-cos3x/2(√3/2)]+5.
=4(sin3x/2*cosπ/3-cos3x/2*sinπ/3)+5.
f(x)=(a-c)^2=4sin(3x/2-π/3)+5.
f(x)的最小正周期T=2π/(3/2)=4π/3.----正最小正周期.
∵sinx,x∈(2kπ-π/2,2kπ+π/2)为增函数,
∴(3x/2-π/3)∈(3kπ-13π/12,3kπ+5π/12) ----是f(x)=4sin(3x/2-π/3)+5的单调增区间；
(3x/2-π/3)∈(3kπ+5π/12,3kπ+23π/12),----是f(x)的单调减区间.
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