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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

A
1
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B
2
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C
3
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D
none of these
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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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