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VECTOR ALGEBRA — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsVECTOR ALGEBRA1 Mark
Question
Let $\vec{\text{a}}$ and $\vec{\text{b}}$ be two unit vectors and a be the angle between them. Then, $\vec{\text{a}}+\vec{\text{b}}$ is a unit vector if:
  1. $\text{a}=\frac{\pi}{4}$
  2. $\text{a}=\frac{\pi}{3}$
  3. $\text{a}=\frac{2\pi}{3}$
  4. $\text{a}=\frac{\pi}{2}$
✓

Answer

  1. $\text{a}=\frac{2\pi}{3}$

Solution:

$\vec{\text{a}}$ and $\vec{\text{b}}$ are unit vectors.

$\Rightarrow|\vec{\text{a}}|=\big|\vec{\text{b}}\big|=1\dots(1)$

Now,

$\vec{\text{a}}.\vec{\text{b}}=|\vec{\text{a}}|\big|\vec{\text{b}}\big|\cos\text{a}$

$\Rightarrow\vec{\text{a}}.\vec{\text{b}}=\cos\text{a}\dots(2)$

[using (1)]

Given that

$\Big|\vec{\text{a}}+\vec{\text{b}}\big|=1$

Squaring both sides, we get

$\big|\vec{\text{a}}+\vec{\text{b}}\big|^2=1$

$\Rightarrow|\vec{\text{a}}|^2+\big|\vec{\text{b}}\big|^2+2\vec{\text{a}}.\vec{\text{b}}=1$

$\Rightarrow1+1+2\cos\text{a}=1$ [using (1) and (2)]

$\Rightarrow2+2\cos\text{a}=1$

$\Rightarrow2\cos\text{a}=-1$

$\Rightarrow\cos\text{a}=\frac{-1}{2}$

$\Rightarrow\text{a}=\frac{2\pi}{3}$

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