If a and b be two unit vectors and the angle between them, than a + b is a unit vector if = 1. 4 2. 3 3. 2 4. 2 3

4. 2 3 Solution: We have | a |=1 and | b | =1 Now, | a + b |=1 | a |^2+ | b | ^2+2 a . b =1 1+1+2| a | | b | =1 2+2 =1 2 =-1 =-1 2 =2 3

If a and b be two unit vectors and the angle between them, than a…

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If $\vec{\text{a}}$ and $\vec{\text{b}}$ be two unit vectors and $\theta$ the angle between them, than $\vec{\text{a}}+\vec{\text{b}}$ is a unit vector if $\theta=$

$\frac{\pi}{4}$
$\frac{\pi}{3}$
$\frac{\pi}{2}$
$\frac{2\pi}{3}$

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Answer

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

Solution:
We have
$|\vec{\text{a}}|=1$ and $\big|\vec{\text{b}\big|}=1$
Now, $\big|\vec{\text{a}}+\vec{\text{b}}\big|=1$
$\Rightarrow|\vec{\text{a}}|^2+\big|\vec{\text{b}\big|}^2+2\vec{\text{a}}.\vec{\text{b}}=1$
$\Rightarrow1+1+2|\vec{\text{a}}|\big|\vec{\text{b}}\big|\cos\theta=1$
$\Rightarrow2+2\cos\theta=1$
$\Rightarrow2\cos\theta=-1$
$\Rightarrow\cos\theta=\frac{-1}{2}$
$\Rightarrow\theta=\frac{2\pi}{3}$

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