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If the magnitudes of two vectors a and $\mathrm{b}$ are equal then which one of the following is correct?
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Verified Answer
The correct answer is:
$(\vec{a}+\vec{b})$ is perpendicular to $(\vec{a}-\vec{b})$
Given $|\overrightarrow{\mathrm{a}}|=|\overrightarrow{\mathrm{b}}|$
Consider $(\vec{a}+\vec{b}) \cdot(\vec{a}-\vec{b})$
$=\vec{a} \cdot \vec{a}-\vec{a} \cdot \vec{b}+\vec{b} \cdot \vec{a}-\vec{b} \cdot \vec{b}$
$=|\overrightarrow{\mathrm{a}}|^{2}-|\overrightarrow{\mathrm{b}}|^{2}=|\overrightarrow{\mathrm{a}}|^{2}-|\overrightarrow{\mathrm{a}}|^{2}=0$
Hence $(\vec{a}+\vec{b})$ is perpendicular to $(\vec{a}-\vec{b})$.
Consider $(\vec{a}+\vec{b}) \cdot(\vec{a}-\vec{b})$
$=\vec{a} \cdot \vec{a}-\vec{a} \cdot \vec{b}+\vec{b} \cdot \vec{a}-\vec{b} \cdot \vec{b}$
$=|\overrightarrow{\mathrm{a}}|^{2}-|\overrightarrow{\mathrm{b}}|^{2}=|\overrightarrow{\mathrm{a}}|^{2}-|\overrightarrow{\mathrm{a}}|^{2}=0$
Hence $(\vec{a}+\vec{b})$ is perpendicular to $(\vec{a}-\vec{b})$.
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