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Scalar Or Dot Product — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsScalar Or Dot Product5 Marks
Question
If $\vec{\text{a}}$ and $\vec{\text{b}}$ are two non-collinear unit vectors such that $\big|\vec{\text{a}}+\vec{\text{b}}\big|=\sqrt{3},$ find $\big(2\vec{\text{a}}-5\vec{\text{b}}\big).\big(3\vec{\text{a}}+\vec{\text{b}}\big).$
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Answer

Given
$\vec{\text{a}}$ and $\vec{\text{b}}$ are unit vectors
then, $|\vec{\text{a}}|=\big|\vec{\text{b}}\big|=1$
$\big|\vec{\text{a}}+\vec{\text{b}}\big|=\sqrt{3}$
Squaring both the sides,
$\big|\vec{\text{a}}+\vec{\text{b}}\big|^2=(\sqrt{3})^2$
$|\vec{\text{a}}|^2+\big|\vec{\text{b}}\big|^2+2\vec{\text{a}}.\vec{\text{b}}=3$
$1+1+2\vec{\text{a}}.\vec{\text{b}}=3$
$2+2\vec{\text{a}}.\vec{\text{b}}=3$
$2\vec{\text{a}}.\vec{\text{b}}=3-2$
$2\vec{\text{a}}.\vec{\text{b}}=1$
$2\vec{\text{a}}.\vec{\text{b}}=\frac{1}{2}$

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