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Scalar Or Dot Product — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsScalar Or Dot Product3 Marks
Question
Find $|\vec{\text{a}|}$ and $\big|\vec{\text{b}}\big|$ if
$\big(\vec{\text{a}}+\vec{\text{b}}\big).\big(\vec{\text{a}}-\vec{\text{b}}\big)=12$ and $|\vec{\text{a}}|=2\big|\vec{\text{b}}\big|$
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Answer

Here, $\big(\vec{\text{a}}+\vec{\text{b}}\big).\big(\vec{\text{a}}-\vec{\text{b}}\big)=12$
$|\vec{\text{a}}|^2-\big|\vec{\text{b}}\big|^2=12$
$\big(2\big|\vec{\text{b}}\big|\big)^2-\big|\vec{\text{b}}\big|^2=12$ $\big[\text{Using|}\vec{\text{a}}|=2\big|\vec{\text{b}}\big|\big]$
$4\big|\vec{\text{b}}\big|^2-\big|\vec{\text{b}}\big|^2=12$
$3\big|\vec{\text{b}}\big|^2=12$
$\big|\vec{\text{b}}\big|^2=\frac{12}{3}$
$\big|\vec{\text{b}}\big|^2=4$
$\big|\vec{\text{b}}\big|=2$
$\big|\vec{\text{a}}\big|=2\big|\vec{\text{b}}\big|=2(2)$
$\big|\vec{\text{a}}\big|=4$
$\big|\vec{\text{b}}\big|=2$

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