1）a·b=1/4(cos3θcosθ-sin3θsinθ)=cos4θ/4
a+b=1/2(cos3θ+cosθ,sin3θ-sinθ)
|a+b|=1/2(2+2cos3θsin3θ-2cosθsinθ)^(1/2)=1/2(2+2cos4θ)^(1/2)=cos2θ
a·b/|a+b|=cos4θ/4cos2θ=(2(cos2θ)^2-1)/4cos2θ
將cos2θ變作自變量,其取值范圍為[1/2,1],a·b/|a+b|是關(guān)于cos2θ的單調(diào)遞增函數(shù),從而a·b/|a+b|的最大值在θ=0時取到,為1/4,最小值在θ=π/3時取到,為-1/4.
2）|ka+b|=（√3）|a-kb|兩邊平方得,并整理得(a^2-3b^2)k^2+8ka·b+b^2-3a^2=0,再將a^2=b^2=1/4及a·b=cos4θ/4代入,得-k^2/2+4kcos4θ-1/2=0,從而k^2-4kcos4θ+1=0,即k+1/k=4cos4θ∈[-4,2],所以,k的取值范圍為[-2-√3,-2+√3]∪\{1}或?qū)懽?2-√3≤k≤-2+√3或k=1