题目内容
设向量a=(3,5,-4),b=(2,1,8),计算2a+3b,3a-2b,a·b及a与b所成的角,并确定λ、μ满足什么关系时,能使λa+μb与z轴垂直.解析:2a+3b=2(3,5,-4)+3(2,1,8)=(6,10,?-8)?+(6,3,24)=(12,13,16).?
3a-2b=3(3,5,-4)-2(2,1,8)=(9,15,-12)-(4,2,16)=(9-4,15-2,-12-16)=(5,13,-28).?
a·b=3×2+5×1-4×8=-21.
|a|=
=
,
|b|=
=
,
∴cos〈a,b〉=
=
=-
=-
.
∴〈a,b〉=arccos(-
).
由(λa+μb)·(0,0,1)=(3λ+2μ,5λ+μ,-4λ+8μ)·(0,0,1)=-4λ+8μ=0,知只要λ、μ满足-4λ+8μ=0,即可使λa+μb与z轴垂直.
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