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Scalar Or Dot Product — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsScalar Or Dot Product3 Marks
Question
If $\vec{\text{a}},\vec{\text{b}}$ are two vectors such that $\big|\vec{\text{a}}+\vec{\text{b}}\big|=\big|\vec{\text{b}}\big|,$ then prove that $\vec{\text{a}}+2\vec{\text{b}}$ is perpendicular to $\vec{\text{a}}.$
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Answer

Given that
$\big|\vec{\text{a}}+\vec{\text{b}}\big|=\big|\vec{\text{b}}\big|$
Squaring both sides, we get
$\big|\vec{\text{a}}+\vec{\text{b}}\big|^2=\big|\vec{\text{b}}\big|^2$
$\Rightarrow|\vec{\text{a}}|^2+\big|\vec{\text{b}}\big|^2+2\vec{\text{a}}.\vec{\text{b}}=\big|\vec{\text{b}}\big|^2$
$\Rightarrow|\vec{\text{a}}|^2+2\vec{\text{a}}.\vec{\text{b}}=0\dots(1)$
Now,
$\big(\vec{\text{a}}+2\vec{\text{b}}\big).\vec{\text{a}}$
$\vec{\text{a}}.\vec{\text{a}}+2\vec{\text{b}}.\vec{\text{a}}$
$=|\vec{\text{a}}|^2+2\vec{\text{a}}.\vec{\text{b}}$
$=0$ [Using (1)]
So, $\vec{\text{a}}+2\vec{\text{b}}$ is perpendicular to $\vec{\text{a}}.$

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