已知向量a=(cosa,sina),向量b=(cosβ,sinβ),|向量a+向量b|=根号下(2+根号2)1)求向量a,b的夹角2)若pi/2<α<pi,-pi/2<β<0,sinβ=-3/5,求cos2α
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![已知向量a=(cosa,sina),向量b=(cosβ,sinβ),|向量a+向量b|=根号下(2+根号2)1)求向量a,b的夹角2)若pi/2<α<pi,-pi/2<β<0,sinβ=-3/5,求cos2α](/uploads/image/z/7018660-28-0.jpg?t=%E5%B7%B2%E7%9F%A5%E5%90%91%E9%87%8Fa%3D%28cosa%2Csina%29%2C%E5%90%91%E9%87%8Fb%3D%28cos%CE%B2%2Csin%CE%B2%29%2C%7C%E5%90%91%E9%87%8Fa%2B%E5%90%91%E9%87%8Fb%7C%3D%E6%A0%B9%E5%8F%B7%E4%B8%8B%282%2B%E6%A0%B9%E5%8F%B72%291%29%E6%B1%82%E5%90%91%E9%87%8Fa%2Cb%E7%9A%84%E5%A4%B9%E8%A7%922%29%E8%8B%A5pi%2F2%EF%BC%9C%CE%B1%EF%BC%9Cpi%2C-pi%2F2%EF%BC%9C%CE%B2%EF%BC%9C0%2Csin%CE%B2%3D-3%2F5%2C%E6%B1%82cos2%CE%B1)
已知向量a=(cosa,sina),向量b=(cosβ,sinβ),|向量a+向量b|=根号下(2+根号2)1)求向量a,b的夹角2)若pi/2<α<pi,-pi/2<β<0,sinβ=-3/5,求cos2α
已知向量a=(cosa,sina),向量b=(cosβ,sinβ),|向量a+向量b|=根号下(2+根号2)
1)求向量a,b的夹角
2)若pi/2<α<pi,-pi/2<β<0,sinβ=-3/5,求cos2α
已知向量a=(cosa,sina),向量b=(cosβ,sinβ),|向量a+向量b|=根号下(2+根号2)1)求向量a,b的夹角2)若pi/2<α<pi,-pi/2<β<0,sinβ=-3/5,求cos2α
1) 向量a+向量b=(cosα+cosβ,sinα+sinβ).
|向量+向量b|=√[(cosα+cosβ)^2+(sinα+sinβ)2]=√(2+√2).
化简,整理,得:cos(α-β)=√2/2.
|向量a|=1,|向量b|=1.
向量a.向量b=cosαcosβ+sinαsinβ.
=cos(α-β)(=√2/2).
cos<a,b>=a.b/|a||b|.
=√2/2.
∴<a,b>=45°---所求向量a,b的夹角.
2) ∵sinβ=-3/5,∴cosβ=4/5.(-π/2<β<0,cosβ>0)
由(1)得:cos(α-β)=√2/2.
∴α-β=45°.
α=45°+β.
cosα=cos(45°+β).
cosα=cos45°cosβ-sin45° sinβ.
=(√2/2)[(4/5)-(-3/5)].
=7√2/10.
cos2α=2cos^2α-1.
=2*(7√2/10)^2-1.
=2*(98/100)-1.
=2*49/50-1.
=(49-25)/25..
∴cos2α=24/25.
1):(a+b)^2=2+√2
a^2+b^2+2ab=2+√2
∵a^2=b^2=1
∴ab=√2/2∴ab夹角为45度
2):好吧,我不会