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Question

Let a and c be unit vectors and |b|=4 with a×b=2a×c. The angle between a and c is cos1(14). If b2c=λa, then λ is equal to
  1. 13,23
  2. 2,3
  3. 3,4
  4. 13,12


Solution

The correct option is C 3,4
It is given that the angle between a and c is cos1(14).
So, ac=|a||c|cos(cos1(14))
ac=14     (1)

b2c=λa
Taking dot products with a, we have
ab2(ac)=λ(aa)
ab12=λ
ab=12+λ

Similarly, taking dot products with b and c
bc=8λ22λ4     (2) 
and bc2=λ(ac)     (3)

From equations (1),(2) and (3), we get
8λ22λ42=λ(14)
λ=3,4
 

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