Question

設(shè)向量a=(3,5,-4),b=(2,1,8),計(jì)算2a+3b,3a-2b,a·b及a與b所成的角,并確定λ、μ滿足什么關(guān)系時,能使λa+μb與z軸垂直.

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=3×2+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滿足-4λ+8μ=0,即可使λa+μb與z軸垂直.