知向量a的膜=根号2,向量b的膜=3,向量a与向量b的夹角为45度.求(1)当向量a+λb与λa+b的夹角为钝角时,λ的取值范围;(2)当λ=-2时,向量a+λb与λa+b的夹角的余弦值.
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![知向量a的膜=根号2,向量b的膜=3,向量a与向量b的夹角为45度.求(1)当向量a+λb与λa+b的夹角为钝角时,λ的取值范围;(2)当λ=-2时,向量a+λb与λa+b的夹角的余弦值.](/uploads/image/z/11564591-23-1.jpg?t=%E7%9F%A5%E5%90%91%E9%87%8Fa%E7%9A%84%E8%86%9C%3D%E6%A0%B9%E5%8F%B72%2C%E5%90%91%E9%87%8Fb%E7%9A%84%E8%86%9C%3D3%2C%E5%90%91%E9%87%8Fa%E4%B8%8E%E5%90%91%E9%87%8Fb%E7%9A%84%E5%A4%B9%E8%A7%92%E4%B8%BA45%E5%BA%A6.%E6%B1%82%EF%BC%881%EF%BC%89%E5%BD%93%E5%90%91%E9%87%8Fa%2B%CE%BBb%E4%B8%8E%CE%BBa%2Bb%E7%9A%84%E5%A4%B9%E8%A7%92%E4%B8%BA%E9%92%9D%E8%A7%92%E6%97%B6%2C%CE%BB%E7%9A%84%E5%8F%96%E5%80%BC%E8%8C%83%E5%9B%B4%EF%BC%9B%EF%BC%882%EF%BC%89%E5%BD%93%CE%BB%3D-2%E6%97%B6%2C%E5%90%91%E9%87%8Fa%2B%CE%BBb%E4%B8%8E%CE%BBa%2Bb%E7%9A%84%E5%A4%B9%E8%A7%92%E7%9A%84%E4%BD%99%E5%BC%A6%E5%80%BC.)
知向量a的膜=根号2,向量b的膜=3,向量a与向量b的夹角为45度.求(1)当向量a+λb与λa+b的夹角为钝角时,λ的取值范围;(2)当λ=-2时,向量a+λb与λa+b的夹角的余弦值.
知向量a的膜=根号2,向量b的膜=3,向量a与向量b的夹角为45度.求(1)当向量a+λb与λa+b的夹角为钝角时,λ的取值范围;(2)当λ=-2时,向量a+λb与λa+b的夹角的余弦值.
知向量a的膜=根号2,向量b的膜=3,向量a与向量b的夹角为45度.求(1)当向量a+λb与λa+b的夹角为钝角时,λ的取值范围;(2)当λ=-2时,向量a+λb与λa+b的夹角的余弦值.
(1)
a²=2,b²=9,
ab=|a||b|cos45°=3
则向量a+λb与λa+b的夹角是钝角时有:
(a+λb)(λa+b)
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(1)∵ab=|a||b|cos45°=3
∴向量a+λb与λa+b的夹角是钝角时有:
(a+λb)(λa+b)<0
λa^2+(1+λ^2)ab+λb^2<0
3λ^2+11λ+3<0
(-11-√85)/6<λ<(-11+√85)/6
向量a+λb与λa+b共线时,对应的系数比相等,
1/λ=1/λ,
λ=±1.
λ=1时...
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(1)∵ab=|a||b|cos45°=3
∴向量a+λb与λa+b的夹角是钝角时有:
(a+λb)(λa+b)<0
λa^2+(1+λ^2)ab+λb^2<0
3λ^2+11λ+3<0
(-11-√85)/6<λ<(-11+√85)/6
向量a+λb与λa+b共线时,对应的系数比相等,
1/λ=1/λ,
λ=±1.
λ=1时,同向共线,夹角为0°,
λ=-1时,反向共线,夹角为180°.
所以(-11-√85)/6<λ<(-11+√85)/6,且λ=-1
(2)
你自己带进去可以了啦
向量a+λb与λa+b的夹角
=cosθ=(a+λb)(λa+b)/|a+λb||λa-b|
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