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.