If for two vector \vec{A} and \vec{B} , sum \(\left(\vec{A} + \vec{B}\right)\) is perpendicular to the difference \(\left(\vec{A} - \vec{B}\right)\) . The ratio of their magnitude is
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\(\left(\vec{A} + \vec{B}\right)\) is perpendicular to \(\left(\vec{A} - \vec{B}\right)\) . Thus
\(\left(\vec{A} + \vec{B}\right)\) . \(\left(\vec{A} - \vec{B}\right)\) = 0
or \(A^{2} + \vec{B} \cdot \vec{A} - \vec{A} \cdot \vec{B} - B^{2} = 0\)
Because of commutative property of dot product \(\vec{A} \cdot \vec{B} = \vec{B} \cdot \vec{A}\)
\dot{-} - - A^{2} - B^{2} = 0 or A = B
Thus the ratio of magnitudes A/B = 1
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