若空间中一向量a与x轴,y轴,z轴正向之夹角依次为α,β,γ则1/(sinα)^2+4/(sinβ)^2+9/(sinγ)^2的最小值

Question

若空间中一向量a与x轴,y轴,z轴正向之夹角依次为α,β,γ则1/(sinα)^2+4/(sinβ)^2+9/(sinγ)^2的最小值

Answer 1

α、β、γ用a、b、c表示：
3个方向余弦满足：cosa^2+cosb^2+cosc^2=1
即：sina^2+sinb^2+sinc^2=2
1/sina^2+4/sinb^2+9/sinc^2
=(1/2)*(2/sina^2)+2*2/sinb^2+(9/2)*(2/sinc^2)
=(1/2)*(1+sinb^2/sina^2+sinc^2/sina^2)+2*(sina^2/sinb^2+1+sinc^2/sinb^2)
+(9/2)*(sina^2/sinc^2+sinb^2/sinc^2+1)
=(1/2+2+9/2)+sinb^2/(2sina^2)+sinc^2/(2sina^2)
+2sina^2/sinb^2+2sinc^2/sinb^2+9sina^2/(2sinc^2)+9sinb^2/(2sinc^2)
≥7+2(sinb*√2sina/(√2sina*sinb))+2(3sinc*sina/(√2sina*√2sinc))
+2(√2sinc*3sinb/(sinb*√2sinc))=18
等号成立的条件：sinb^2=2sina^2,sinc^2=3sina^2,2sinc^2=3sinb^2
但这个等号成立的条件比较苛刻,还要考虑sina^2+sinb^2+sinc^2=2的条件
即：6sina^2=2,即：sina^2=1/3,sinb^2=2/3,sinc^2=1
即：sina=√3/3,sinb=√6/3,sinc=1
此时1/sina^2+4/sinb^2+9/sinc^2取得最小值：18