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Question

For two vectors A and B, sum A+B is perpendicular to the difference AB. Then the ratio of their magnitudes may be 
  1. 1
  2. 2
  3. 3
  4. none of these 


Solution

The correct option is A 1
(A+B) is perpendicular to (AB).
Thus (A+B)(AB)=0
AAAB+BABB=0
By commutative property of dot product
A.B=B.A
 A2B2=0 or A=±B

Thus the ratio of magnitudes may be
AB=1 (since magnitude cannot be negative)

Alternative: According to parallelogram law of vector addition, A+B and AB are diagonals.
Because diagonals are perpendicular according to question, it is a rhombus. Hence the ratio of magnitudes of the two vectors is 1.

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