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VECTOR ALGEBRA — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsVECTOR ALGEBRA1 Mark
Question
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 $\theta=$
  1. $\frac{\pi}{4}$
  2. $\frac{\pi}{3}$
  3. $\frac{\pi}{2}$
  4. $\frac{2\pi}{3}$
✓

Answer

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

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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