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Vector or Cross Product — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSVector or Cross Product3 Marks
Question
If $|\vec{\text{a}}|=\sqrt{26,}\big|\vec{\text{b}}\big|=7$ and $\big|\vec{\text{a}}\times\vec{\text{b}}\big|=35,$ find $\vec{\text{a}}.\vec{\text{b}}.$
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Answer

$\vec{\text{a}}\times\vec{\text{b}}=|\vec{\text{a}}|\big|\vec{\text{b}}\big|\sin\theta.\hat{\text{n}}$
$\big|\vec{\text{a}}\times\vec{\text{b}}\big|=|\vec{\text{a}}|\big|\vec{\text{b}}\big||\sin\theta||\hat{\text{n}}|$
$35=\sqrt{26.7}|\sin\theta|.1$
$\sin\theta=\frac{35}{\sqrt{26.5}}$
$\sin\theta=\frac{5}{\sqrt{26}}$
$\cos^2\theta=1-\sin^2\theta$
$=1-\Big(\frac{5}{\sqrt{26}}\Big)^2$
$=\frac{1}{1}-\frac{25}{26}$
$=\frac{26-25}{26}$
$=\frac{1}{26}$
$\cos\theta=\frac{1}{\sqrt{26}}$
$\vec{\text{a}}.\vec{\text{b}}=|\vec{\text{a}}|\big|\vec{\text{b}}\big|\cos\theta$
$=\sqrt{26}.7.\frac{1}{\sqrt{26}}$
$\vec{\text{a}}.\vec{\text{b}}=7$

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