For two vectors →A and →B, sum →A+→B is perpendicular to the difference →A−→B. 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 (→A−→B).
Thus (→A+→B)⋅(→A−→B)=0 →A⋅→A−→A⋅→B+→B⋅→A−→B⋅→B=0
By commutative property of dot product →A.→B=→B.→A ∴A2−B2=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 →A−→B are diagonals.
Because diagonals are perpendicular according to question, it is a rhombus. Hence the ratio of magnitudes of the two vectors is 1.