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STD 12 - 10. vector algebra — Mathematics STD 12 Science — Question

CBSE BoardEnglish MediumSTD 12 ScienceMathematicsSTD 12 - 10. vector algebra1 Mark
MCQ
If $2\vec a + 3\vec b + \vec c = \vec 0$, then  $\vec a \times \vec b + \vec b \times \vec c + \vec c \times \vec a$ is equal to
  • A
    $6\left( {\vec b \times \vec c} \right)$
  • ✓
    $3\left( {\vec b \times \vec c} \right)$
  • C
    $2\left( {\vec b \times \vec c} \right)$
  • D
    $\vec 0$
✓

Answer

Correct option: B.
$3\left( {\vec b \times \vec c} \right)$
b
Given, $2\overrightarrow a  + 3\overrightarrow b  + \overrightarrow c  = \overrightarrow 0 $

$ \Rightarrow 2\overrightarrow a  + 3\overrightarrow b  =  - \overrightarrow c $

Taking cross product with $\overrightarrow{\mathrm{a}}$ and $\overrightarrow{\mathrm{b}}$ respectively, we get

$2(\overrightarrow a  \times \overrightarrow a ) + 3(\overrightarrow a  \times \overrightarrow b ) =  - \overrightarrow a  \times \overrightarrow c $

$ \Rightarrow \quad 3(\overrightarrow a  \times \overrightarrow b ) = \overrightarrow c  \times \overrightarrow a $      .......$(i)$

and $\quad 2(\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{a}})+3(\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{b}})=-\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}$

$ \Rightarrow \quad 2(\overrightarrow a  \times \overrightarrow b ) = \overrightarrow b  \times \overrightarrow c $      .....$(ii)$

Now, $\vec{a} \times \vec{b}+\vec{b} \times \vec{c}+\vec{c} \times \vec{a}$

$ = \overrightarrow a  \times \overrightarrow b  + \overrightarrow b  \times \overrightarrow c  + 3(\overrightarrow a  \times \overrightarrow b )\quad $      [using Eq. $(i)$] 

$ = 4(\overrightarrow a  \times \overrightarrow b ) + \overrightarrow b  \times \overrightarrow c $

$ = 2(\overrightarrow b  \times \overrightarrow c ) + \overrightarrow b  \times \overrightarrow c $           [using Eq. $(ii)$] 

$ = 3(\overrightarrow b  \times \overrightarrow c )$

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