If a, b and (a+b) are all unit vectors and is the angle between a… - Vidyadip

MCQ

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 :

✓
$\frac{2 \pi}{3}$
B
$\frac{5 \pi}{6}$
C
$\frac{\pi}{3}$
D
$\frac{\pi}{6}$

✓

Answer

Correct option: A.

$\frac{2 \pi}{3}$

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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