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Question

Let a=2^i+^j^k and b=^i+2^j+^k be two vectors. Consider a vector c=αa+βb, α,βR. If the projection of c on the vector (a+b) is 32, then the minimum value of (c(a×b)).c equals

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Solution

Let c=αa+βb=(2α+β)^i+(α+2β)^j+(βα)^k
Projection c on vector a+b
=c.(a+b)|a+b|=32
((2α+β)^i+(α+2β)^j+(βα)^k).(3^i+3^j+0^k)|3^i+3^j+0^k|=32
9(α+β)=32×32
α+β=2...(i)
(a×b)=∣ ∣ ∣^i^j^k211121∣ ∣ ∣=3^i3^j+3^k
(ca×b).c=|c|2[abc]
[abc]=0
|c|2=(2α+β)2+(α+2β)2+(βα)2=6(α2+β2+α.β)
For minimum value α=β=1
Hence,(ca×b).c=18

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