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Vector Algebra — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSVector Algebra1 Mark
Question
If $\vec{a}, \vec{b}$ and $(\vec{a}+\vec{b})$ are all unit vectors and $\theta$ is the angle between $\vec{a}$ and $\vec{b}$, then the value of $\theta$ is :
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Answer

Given, $|\vec{a}|=|\vec{b}|=|\vec{a}+\vec{b}|=1$
Now, $|\vec{a}+\vec{b}|^2=|\vec{a}|^2+|\vec{b}|^2+2|\vec{a}||\vec{b}| \cos \theta$
$\Rightarrow 1^2=1^2+1^2+2 \cdot 1 \cdot 1 \cos \theta$
$\Rightarrow \cos \theta=\frac{-1}{2} \Rightarrow \theta=\cos ^{-1}\left(\frac{-1}{2}\right)$
$\Rightarrow \theta=\frac{2 \pi}{3}$

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