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Question

Let A be the set of vectors a=(a1,a2,a3) satisfying (3i=1ai2i)2=3i=1a2i2i. Then


Your Answer
A
A is empty
Correct Answer
B
A contains exactly one element
Your Answer
C
A has 6 elements
Your Answer
D
A has infinitely many elements 

Solution

The correct option is B A contains exactly one element
Let p=(a12,a24,a38) & q=(12,14,18) and angle between them is θ

Given that (3i=1ai2i)2=3i=1a2i2i.

78(3i=1ai2i)2=783i=1a2i2i

78(a12+a24+a38)2=78(a212+a224+a238)78(pq)2=|p|2|q|278|p|2|q|2 cos2θ=|p|2|q|2|p|=0  or  cos2θ=87

cos2θ=87 is not possible. |p|=0p=(0,0,0)a=(0,0,0)
Therefore, the equation has only one solution.

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