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Question

If the pth,qth and rth terms of a G.P are positive numbers a,b and c respectively, then find the angle between the vectors loga3^i+logb3^j+logc3^k and (qr)^i+(rp)^j+(pq)^k

A
π6
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B
π4
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C
π3
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D
π2
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Solution

The correct option is C π2
Accordingtoquestion.................pthterm:a=mnp1suppose:firstterm=mcommonratioofG.P=nb=mnq1c=mnr1firstvector:3logaˆi+3logbˆj+3logcˆkx=3(logm+(p1)logn)ˆi+3(logm+(q1)logn)ˆj+3(logm+(r1)logn)ˆkand,y=(qr)ˆi+(rp)ˆj+(pq)ˆkdotprodut:x.y=3(logm.qr)+3logn[(p1)(qr)]+3(logm.rp)+3logn[(q1)(rp)]++3(logm.pq)+3logn[(r1)(pq)]=3logm(qr+rp+pq)+3logn[(p1)(qr)+(q1)(rp)+(r1)(pq)]=3logm(qr+rp+pq)+3logn[pqprq+r+qrqpr+p+rprqp+q]=3logm(0)+3logn[0]=0+0=0x.y=0x.y=|x||y|cosθcosθ=0θ=π2so,thatthecorrectoptionisD.

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