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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
Let $\vec a = \hat i + \hat j + \hat k,\,\,\,\vec c = \hat j - \hat k$ and a vector $\vec b$ be such that $\vec a \times \vec b = \,\vec c$ and $\vec a\, \cdot \,\vec b = \,3.$ Then $\left| {\vec b} \right|$ equals?
  • ✓
    $\sqrt {\frac{{11}}{3}} $
  • B
    $\frac{{\sqrt {11} }}{3}$
  • C
    $\frac{{11}}{{\sqrt 3 }}$
  • D
    $\frac {11}{3}$
✓

Answer

Correct option: A.
$\sqrt {\frac{{11}}{3}} $
a
$\because \vec{a}=\hat{i}+\hat{j}+\hat{k} \Rightarrow|\vec{a}|=\sqrt{3}$

$\vec{c}=\hat{j}-\hat{k} \Rightarrow(\text { Given })|\bar{c}| \sqrt{2}$

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

$\Rightarrow|\vec{a}||\vec{b}| \sin \theta=|\vec{c}|$

$\Rightarrow|\vec{a}||\vec{b}| \sin \theta=\sqrt{2}$          ........$[i]$

Also $\vec{a} \cdot \vec{b}=3$

$\Rightarrow|\vec{a}||\vec{b}| \cos \theta=3 $        ........$[ii]$

Dividing $[i]$ by $[ii],$ we get

$\tan \theta=\frac{\sqrt{2}}{3}$

$\therefore \sin \theta=\frac{\sqrt{2}}{\sqrt{11}}$

Substituting value of $\sin \theta$ in $[i]$ we get

$\sqrt{3}|\vec{b}| \frac{\sqrt{2}}{\sqrt{11}}=\sqrt{2}$

$|\vec{b}|=\frac{\sqrt{11}}{\sqrt{3}}$

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