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Question

a and b are two vectors such that |a|=1, |b|=4, |c|2=192 and a.b=2. If c=(2a×b)3b, then the angle between b and c is

A
π3
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B
π6
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C
3π4
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D
5π6
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Solution

The correct option is D 5π6
|a|=1, |b|=4, ab=2
c=(2a×b)3b
c+3b=2a×b

ab=2
|a||b|cosθ=2
cosθ=2|a||b|=24=12
θ=π3

|c+3b|2=|2a×b|2
|c|2+9|b|2+2c3b=4|a|2|b|2sin2θ
|c|2+9|b|2+6bc=48
|c|2+96+6(bc)=0 [1]
c=2a×b3b
bc=03×16=48

Putting value of bc in [1], we have
|c|2+966×48=0|c|=83
Putting this value in [1], we get
192+96+6|b||c|cosα=0
cosα=32
or, α=5π6

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