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VECTOR ALGEBRA — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSVECTOR ALGEBRA1 Mark
Question
State True or False for the following:
If $|\vec{\text{a}}+\vec{\text{b}}|=|\vec{\text{a}}-\vec{\text{b}}|,$ then the vector $\vec{\text{a}}$ and $\vec{\text{b}}$ are orthogonal.
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Answer

True.Solution:
Given, $|\vec{\text{a}}+\vec{\text{b}}|=|\vec{\text{a}}-\vec{\text{b}}|$
$\Rightarrow|\vec{\text{a}}+\vec{\text{b}}|^2=|\vec{\text{a}}-\vec{\text{b}}|^2$
$\Rightarrow|\vec{\text{a}}|^2+|\vec{\text{b}}|^2+2\vec{\text{a}}\cdot\vec{\text{b}}=|\vec{\text{a}}|^2+|\vec{\text{b}}|^2-2\vec{\text{a}}\cdot\vec{\text{b}}$
$\Rightarrow2\vec{\text{a}}\cdot\vec{\text{b}}=-2\vec{\text{a}}\cdot\vec{\text{b}}$
$\Rightarrow4\vec{\text{a}}\cdot\vec{\text{b}}=0$
$\Rightarrow\vec{\text{a}}\cdot\vec{\text{b}}=0$
Hence, $\vec{\text{a}}$ and $\vec{\text{b}}$ are orthogonal.

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