若向量a=(2,1)圍繞原點逆時針旋轉π/4,得到向量b,則b的坐標是多少?

熱心網友

設向量a=(2,1)的傾角為α,向量b的傾角為β則sinα=y/r=1/√5 cosα=x/r=2/√5 sinβ=sin(α+π/4)=sinαcos(π/4)+cosαsin(π/4) =(1/√5)*(√2/2)+(2/√5)*(√2/2) =(3√2)/(2√5) cosβ=cos(α+π/4)=cosαcos(π/4)-sinαsin(π/4) =(2/√5)*(√2/2)-(1/√5)*(√2/2) =√2/(2√5) r=√5向量b的橫坐標為 cosβ*r=√2/(2√5)*√5=√2/2 縱坐標為 sinβ*r=(3√2)/(2√5)*√5=3√2/2 向量b=(√2/2,3√2/2)

熱心網友

解：設a=（OA→）=2+i，b=(OB→)，由題意結合復數的幾何意義得(OB→) =（OA→） (cosπ/4+isinπ/4)=(2+i)(√2/2)+ [(√2/2)i]= (√2/2)+ [(3√2/2)i], 即b= (√2/2, 3√2/2)