Question

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

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

Solution

## The correct option is B A contains exactly one elementLet →p=(a1√2,a2√4,a3√8) & →q=(1√2,1√4,1√8) and angle between them is θ Given that (3∑i=1ai2i)2=3∑i=1a2i2i. ⇒78(3∑i=1ai2i)2=783∑i=1a2i2i ⇒78(a12+a24+a38)2=78(a212+a224+a238)⇒78(→p⋅→q)2=|→p|2|→q|2⇒78|→p|2|→q|2 cos2θ=|→p|2|→q|2⇒|→p|=0  or  cos2θ=87 cos2θ=87 is not possible. ⇒|→p|=0⇒→p=(0,0,0)⇒→a=(0,0,0) Therefore, the equation has only one solution.

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