We have, v1,v2,v3,v4 be unit vectors lie in XY-plane. Let v1=cosαi+sinαjα∈(0,90∘) v2=cosβi+sinβjβ∈(90∘,180∘) v3=cosγi+sinγjγ∈(180∘,270∘) v1=cosδi+sinδjδ∈(270∘,360∘)
(a) v1+v2+v3+v4=0 not necessarily true for all v1,v2,v3 and v4
(b) $v_{i}+v_{j}(1 ≤ iv1+v2=(cosα+cosβ)i+(sinα+sinβ)j If y coordinate is positive i.e. α in 1 st quadrant β in 2 nd quadrant. But in this case x-coordinate is not necessarily positive.
(c) $ \;\; v_{i}=v_{j}(1 ≤ i Let α in 1 st quadrant, γ in 3 rd quadrant ∴vi⋅vj<0
(d) vi⋅vj=(cosαcosβ)+sinαsinβ=cos(α−β) is positive 0<|α−β|<
π
2
=
3π
2
<|α−β|<2π Not positive for all values of v1,v2,v3,v4.