Question

已知向量a=（2cosφ,2sinφ),φ∈（π/2,π）,向量b=（0,-1）,则向量a与b的夹角为

Answer 1

【1】可设两个向量a,b的夹角为θ,易知：0º≤θ≤180º.【2】∵向量a=(2cosψ,2sinψ).向量b=(0,-1).∴|a|=2,|b|=1.且a·b=-2sinψ.又cosθ=(a·b)/(|a|×|b|).∴cosθ=(-2sinψ)/2=-sinψ.即有：cosθ=-sinψ.【3】由题设可知：ψ∈(π/2,π),即π/2＜ψ＜π.∴0＜sinψ＜1.∴-1＜-sinψ＜0.∴-1＜cosθ＜0.结合0≤θ≤π,可知：π/2＜θ＜π.【4】∵cosθ=-sinψ.∴cos(π-θ)=sinψ=cos[(π/2)-ψ]=cos[ψ-(π/2)].即有：cos(π-θ)=cos[ψ-(π/2)].由π/2＜θ＜π,且π/2＜ψ＜π可知：0＜π-θ＜π/2.且0＜ψ-（π/2)＜π/2.∴π-θ=ψ-(π/2).∴θ=(3π/2)-ψ.即两个向量a,b的夹角为(3π/2)-ψ.∴选A.