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Scalar Or Dot Product — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsScalar Or Dot Product3 Marks
Question
If $\vec{\text{c}}$ is perpendicular to both $\vec{\text{a}}$ and $\vec{\text{b}},$ then prove that it is perpendicular to both $\vec{\text{a}}+\vec{\text{b}}$ and $\vec{\text{a}}-\vec{\text{b}}.$
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Answer

Given that $\vec{\text{c}}$ is perpendicular to both $\vec{\text{a}}$ and $\vec{\text{b}}.$
$\Rightarrow\vec{\text{c}}.\vec{\text{a}}=0$ and $\vec{\text{c}}.\vec{\text{b}}=0\dots(1)$
Now,
$\vec{\text{c}}.\big(\vec{\text{a}}+\vec{\text{b}}\big)=\vec{\text{c}}.\vec{\text{a}}+\vec{\text{c}}.\vec{\text{b}}=0+0=0$ [From (1)]
So, $\vec{\text{c}}$ is perpendicular to $\vec{\text{a}}+\vec{\text{b}}.$
Again,
$\vec{\text{c}}.\big(\vec{\text{a}}-\vec{\text{b}}\big)=\vec{\text{c}}.\vec{\text{a}}-\vec{\text{c}}.\vec{\text{b}}=0-0=0$  [From (1)]
So, $\vec{\text{c}}$ is perpendicular to $\vec{\text{a}}-\vec{\text{b}}.$

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