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VECTOR ALGEBRA — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsVECTOR ALGEBRA1 Mark
Question
If $\vec{\text{a}}.\vec{\text{b}}=\vec{\text{a}}.\vec{\text{c}}$ and $\vec{\text{a}}\times\vec{\text{b}}=\vec{\text{a}}\times\vec{\text{c}}.\vec{\text{a}}\neq0,$ then:
  1. $\vec{\text{b}}=\vec{\text{c}}$
  2. $\vec{\text{b}}=\vec{0}$
  3. $\vec{\text{b}}+\vec{\text{c}}=\vec{0}$
  4. $\text{None of these}$
✓

Answer

  1. $\vec{\text{b}}=\vec{\text{c}}$

Solution:

$\vec{\text{a}}.\vec{\text{b}}=\vec{\text{a}}.\vec{\text{c}}$

$\Rightarrow\vec{\text{a}}.\vec{\text{b}}-\vec{\text{a}}.\vec{\text{c}}=0$

$\Rightarrow\vec{\text{a}}.\big(\vec{\text{b}}-\vec{\text{c}}\big)=0$

Let $\theta$ be the angle between $\vec{\text{a}}$ and $\big(\vec{\text{b}}-\vec{\text{c}}\big)$

$|\vec{\text{a}}|\big|\big(\vec{\text{b}}-\vec{\text{c}}\big)\big|\cos\theta\dots(1)$

and $\vec{\text{a}}\times\vec{\text{b}}=\vec{\text{a}}\times\vec{\text{c}}$

$\Rightarrow\vec{\text{a}}\times\vec{\text{b}}-\vec{\text{a}}\times\vec{\text{c}}=0$

$\Rightarrow\vec{\text{a}}\times\big(\vec{\text{b}}-\vec{\text{c}}\big)=0$

Then, $|\vec{\text{a}}|\big|\big(\vec{\text{b}}-\vec{\text{c}}\big)\big|\sin\theta=0\dots(2)$

Here, it is given that $\vec{\text{a}}\neq0$

Therefore, for eq. (1) and eq. (2) to be 0

We have,

$\big|\big(\vec{\text{b}}-\vec{\text{c}}\big)\big|\cos\theta=0$

For $\big|\big(\vec{\text{b}}-\vec{\text{c}}\big)\big|\cos\theta=0,$ one of $\big|\big(\vec{\text{b}}-\vec{\text{c}}\big)\big|$ or $\cos\theta$ must be 0

Case 1 :

Let $\cos\theta=0$

$\Rightarrow\theta=90^\circ$

$\Rightarrow\sin\theta=1$

& if $\big|\big(\vec{\text{b}}-\vec{\text{c}}\big)\big|\sin\theta=0$ and $\sin\theta=1$

Then $\big|\big(\vec{\text{b}}-\vec{\text{c}}\big)\big|=0$

$\Rightarrow\vec{\text{b}}=\vec{\text{c}}$

Case 2 :

Let $\big|\big(\vec{\text{b}}-\vec{\text{c}}\big)\big|=0$

$\Rightarrow\vec{\text{b}}=\vec{\text{c}}$

Hence, $\vec{\text{b}}=\vec{\text{c}}$

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