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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 $\vec a = 3\hat i + 2\hat j + x\hat k$ and $\vec b = \hat i - \hat j + \hat k$, for some real $x$. Then $\left| {\vec a \times \vec b} \right| = r$ is possible if
  • ✓
    $r \geq 5\sqrt {\frac{3}{2}} $
  • B
    $3\sqrt {\frac{3}{2}}  < r < 5\sqrt {\frac{3}{2}} $
  • C
    $\sqrt {\frac{3}{2}}  < r \leq 3\sqrt {\frac{3}{2}} $
  • D
    $0 < r \leq \sqrt {\frac{3}{2}} $
✓

Answer

Correct option: A.
$r \geq 5\sqrt {\frac{3}{2}} $
a
$\vec a \times \vec b = \left| {\begin{array}{*{20}{c}}
{\hat i}&{\hat j}&{\hat k}\\
3&2&x\\
1&{ - 1}&1
\end{array}} \right|$

$=(2+x) \hat{i}-(3-x) \hat{j}-5 \hat{k}$

$|\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}|=\sqrt{2\left(\mathrm{x}^{2}-\mathrm{x}+19\right)}$

$=\sqrt{2} \sqrt{(x-1 / 2)^{2}+19^{-14}} \geq \frac{5 \sqrt{3}}{\sqrt{2}}$

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