设空间两个单位向量向量OA=(m,n,0),向量OB=(0,n,p)与向量OC=(1,1,1)的夹角都等于π/4,
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![设空间两个单位向量向量OA=(m,n,0),向量OB=(0,n,p)与向量OC=(1,1,1)的夹角都等于π/4,](/uploads/image/z/10196679-39-9.jpg?t=%E8%AE%BE%E7%A9%BA%E9%97%B4%E4%B8%A4%E4%B8%AA%E5%8D%95%E4%BD%8D%E5%90%91%E9%87%8F%E5%90%91%E9%87%8FOA%3D%EF%BC%88m%2Cn%2C0%29%2C%E5%90%91%E9%87%8FOB%3D%EF%BC%880%2Cn%2Cp%29%E4%B8%8E%E5%90%91%E9%87%8FOC%3D%281%2C1%2C1%EF%BC%89%E7%9A%84%E5%A4%B9%E8%A7%92%E9%83%BD%E7%AD%89%E4%BA%8E%CF%80%EF%BC%8F4%2C)
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设空间两个单位向量向量OA=(m,n,0),向量OB=(0,n,p)与向量OC=(1,1,1)的夹角都等于π/4,
设空间两个单位向量向量OA=(m,n,0),向量OB=(0,n,p)与向量OC=(1,1,1)的夹角都等于π/4,
设空间两个单位向量向量OA=(m,n,0),向量OB=(0,n,p)与向量OC=(1,1,1)的夹角都等于π/4,
|OA|=1
=> m^2+n^2 = 1 (1)
OA.OC = |OA||OC|cos(π/4)
(m,n,0).(1,1,1) = 1. √3 ( √2/2)
m+n = √6/2 (2)
sub (2) into (1)
m^2 +(√6/2-m)^2 =1
4m^2 - 2√6m + 1 =0
m = (√6 +√2)/4 or (√6 -√2)/4
when m= (√6 +√2)/4 , n= (√6 -√2)/4
when m= (√6 -√2)/4 , n= (√6 +√2)/4
OA = ((√6 +√2)/4 , (√6 -√2)/4, 0) or ((√6 -√2)/4 ,(√6 +√2)/4,0)
Similarly,
OB.OC = |OB||OC|cos(π/4)
(0,n,p). (1,1,1) = 1. √3 (√2/2)
n+p = √6/2
p = √6/2 - n
when n = (√6 -√2)/4, p=(√6 +√2)/4
when n = (√6 +√2)/4, p=(√6 -√2)/4
OB =(0, (√6 -√2)/4, (√6 +√2)/4) or (0, (√6 +√2)/4, (√6 -√2)/4)