First slide

Vectors

Question

If for two vector $\vec{A} and \vec{B}$ , sum ( $\vec{A} + \vec{B}$ ) is perpendicular to the difference ( $\vec{A} - \vec{B}$ ). The ratio of their magnitude is

Moderate

A

1
B

2
C

3
D

None of these

Solution

$(\vec{A} + \vec{B}) is perpendicular to (\vec{A} - \vec{B}) .$ Thus
$(\vec{A} + \vec{B}) . (\vec{A} - \vec{B}) = 0$
or $A^{2} + \vec{B} . \vec{A} - \vec{A} . \vec{B} - B^{2} = 0$
Because of commutative property of dot product
$\vec{A} . \vec{B} = \vec{B} . \vec{A}$
$∴ A^{2} - B^{2} = 0 or A = B$
Thus the ratio of magnitudes $\frac{A}{B} = 1$

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