a and b are unit vectors. If a+b is a unit vector, then the angle… - Vidyadip

MCQ

$\bar{a}$ and $\bar{b}$ are unit vectors. If $\bar{a}+\bar{b}$ is a unit vector, then the angle between $\overline{ a }$ and $\overline{ b }$ is

A
$\frac{\pi}{4}$
B
$\frac{\pi}{3}$
✓
$\frac{2 \pi}{3}$
D
$\frac{3 \pi}{2}$

✓

Answer

Correct option: C.

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

(C) $|\overline{ a }+\overline{ b }|=1$
$\Rightarrow|\overline{ a }+\overline{ b }|^2=1$
$\Rightarrow(\overline{ a }+\overline{ b }) \cdot(\overline{ a }+\overline{ b })=1 \\ \Rightarrow \overline{ a } \cdot \overline{ a }+2 \overline{ a } \cdot \overline{ b }+\overline{ b } \cdot \overline{ b }=1 \\ \Rightarrow 1+2 ab \cos \theta+1=1 \\ \Rightarrow 2 \times 1 \times 1 \times \cos \theta=-1 \\ \Rightarrow \cos \theta=-\frac{1}{2} \\ \Rightarrow \theta=\frac{2 \pi}{3}$

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