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Scalar Or Dot Product — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsScalar Or Dot Product2 Marks
Question
Find the angle between the vectors $\vec{\text{a}} $ and $\vec{\text{b}},$ where$\vec{\text{a}}=2\hat {\text{i}}-3\hat{\text{j}}+\hat{\text{k}}$ and $\vec{\text{b}} =\hat{\text{i}}+\hat{\text{j}}-2\hat{\text{k}}$
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Answer

Let $\theta$ be the angle between $\vec{\text{a}}$ and $\vec{\text{b}}$.$\big|\vec{\text{a}}\big|=\sqrt{(2)^2+(-3)^2+(1)^{2}}=\sqrt{14}$
$\big|\vec{\text{b}}\big|=\sqrt{(1)^2+(1)^2+(-2)^{2}}=\sqrt{6}$
$\vec{\text{a}}.\vec{\text{b}}=2-3-2=-3$
$\cos\theta=\frac{\vec{\text{a }}.\vec{\text{ b}}}{\big|\vec{a}\big|\big|\vec{b}\big|}=\frac{-3}{\sqrt{14}\sqrt{6}}=\frac{-3}{\sqrt{84}}$
$\Rightarrow\theta=\cos^{-1}\Big(\frac{-3}{\sqrt{84}}\Big)$

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