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Question

If a unit vector ^a, makes angles π3 with ^i,π4 with ^j and an acute angle θ with ^k, then find θ and hence the components of a.

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Solution

Let unit vector makes angle α,β and γ with ^i,^j and ^k respectively, then
α=π3,β=π4 and γ=θ (Given)
cos2 π3+cos2 π4+cos2 θ=1(12)2+(12)2+cos2 θ=114+12+cos2 θ=1cos2 θ=134cos2 θ=434=14cos θ=±14cos θ=±12
cos θ=12 [cos θ12,θ is an acute angle]
θ=cos1(12)=cos1(cos π3)
θ=π3 and components of a are cos π3, cos π4, cos π3
12,12,12
Note For acute angle, cos θ>0 and for obtuse angle cos θ<0


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