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If $|\vec{a}+\vec{b}|=|\vec{a}-\vec{b}|$,then which one of the following is correct ?
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Verified Answer
The correct answer is:
$\vec{a}$ is perpendicular to $\vec{b}$.
Since, $|a+b|=|a-b|$ $\Rightarrow[|a+b|]^{2}=[a-b]^{2}$
$\Rightarrow a \cdot a+b \cdot b+a \cdot b+b \cdot a=a \cdot a+b \cdot b-a \cdot b-b \cdot a$
$\Rightarrow 4 a \cdot b=0 \quad(\because a \cdot b=b . a)$
$\Rightarrow a \cdot b=0$
Hence, a is perpendicular to $\mathrm{b}$.
$\Rightarrow a \cdot a+b \cdot b+a \cdot b+b \cdot a=a \cdot a+b \cdot b-a \cdot b-b \cdot a$
$\Rightarrow 4 a \cdot b=0 \quad(\because a \cdot b=b . a)$
$\Rightarrow a \cdot b=0$
Hence, a is perpendicular to $\mathrm{b}$.
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