求与向量a=(7/2,1/2),b=(1/2,7/2)的夹角相等,且模长为1的向量

问题描述:

求与向量a=(7/2,1/2),b=(1/2,7/2)的夹角相等,且模长为1的向量

选择题吧 题干错吧

设所求向量c=(m,n),
|c|=√(m^2+n^2)=1,
设向量a和c夹角为θ
cosθ=a·c/(|a||c|=(7m/2+n/2)/[√(49/4+1/4)*1]=√2(7m/2+n/2)/5,
cosθ=b·c(/|b||c|)=(m/2+7n/2)/√[(1/4+49/4)*1]=√2(m/2+7n/2)/5,
√2(7m/2+n/2)/5=√2(m/2+7n/2)/5,
m=n,
m^2+n^2=1,
m=±√2/2,
n=±√2/2,
m,n应取同号
则向量c=(√2/2,√2/2), c=(-√2/2,-√2/2),

设所求向量c=(m,n),
|c|=√(m^2+n^2)=1,
设向量a和c夹角为θ
cosθ=a·c/(|a||c|=(7m/2+n/2)/[√(49/4+1/4)*1]=√2(7m/2+n/2)/5,
cosθ=b·c(/|b||c|)=(m/2+7n/2)/√[(1/4+49/4)*1]=√2(m/2+7n/2)/5,
√2(7m/2+n/2)/5=√2(m/2+7n/2)/5,
m=n,
m^2+n^2=1,
m=±√2/2,
n=±√2/2,
m,n应取同号
则向量c=(√2/2,√2/2),c=(-√2/2,-√2/2),