Question

设向量a=(3,5，-4)，b=（2,1，8），计算2a+3b,3a-2b,a·b，并确定λ、μ的关系，使λa+μb与z轴垂直.

Answer 1

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=(3,5,-4)·(2,1,8)=3×2+5×1-4×8=-21.

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λ-2μ=0即可使λa+μb与z轴垂直.