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Scalar Or Dot Product — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsScalar Or Dot Product3 Marks
Question
Find $|\vec{\text{a}|}$ and $\big|\vec{\text{b}}\big|$ if
$\big(\vec{\text{a}}+\vec{\text{b}}\big).\big(\vec{\text{a}}-\vec{\text{b}}\big)=3$ and $|\vec{\text{a}}|=2\big|\vec{\text{b}}\big|$
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Answer

Here, $\big(\vec{\text{a}}+\vec{\text{b}}\big).\big(\vec{\text{a}}-\vec{\text{b}}\big)=3$ $|\vec{\text{a}}|^2-\big|\vec{\text{b}}\big|^2=3$ $\big(2\big|\vec{\text{b}}\big|\big)^2-\big|\vec{\text{b}}\big|^2=3$ $\big[\text{using} |\vec{\text{a}}|=2\big|\vec{\text{b}}\big|\big]$ $4\big|\vec{\text{b}}\big|^2-\big|\vec{\text{b}}\big|^2=3$$3\big|\vec{\text{b}}\big|^2=3$
$\big|\vec{\text{b}}\big|^2=\frac{3}{3}$
$\big|\vec{\text{b}}\big|^2=1$
$\big|\vec{\text{b}}\big|=1$
$|\vec{\text{a}}|=2\big|\vec{\text{b}}\big|$ $=2(1)$ $|\vec{\text{a}}|=2$ $\big|\vec{\text{b}}\big|=1$

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