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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 three points $A(2,3,4), B(3,4,2)$ and $C(4,2,3)$ in space are given. A point $D$ in space is such that it is at a distance of $\sqrt 6$ units from $3$ given points. Then volume of tetrahedron $ABCD$ is -
  • A
    $1$
  • ✓
    $\sqrt 3 $
  • C
    $\sqrt 13 $
  • D
    $2$
✓

Answer

Correct option: B.
$\sqrt 3 $
b
$A B=B C=C A=\sqrt{6}$

It is a regular tetrahedron

$\left[ {\vec a\,\vec b\,\vec c} \right]{\,^2} = \left| {\begin{array}{*{20}{c}}
{\vec a \cdot \vec a}&{\vec a \cdot \vec b}&{\vec a \cdot \vec c}\\
{\vec b \cdot \vec a}&{\vec b \cdot \vec b}&{\vec b \cdot \vec c}\\
{\vec c \cdot \vec a}&{\vec c \cdot \vec b}&{\vec c \cdot \vec c}
\end{array}} \right| = \left| {\begin{array}{*{20}{c}}
6&3&3\\
3&6&3\\
3&3&6
\end{array}} \right|$

$[a b c]^{2}=108$

volume $=\frac{1}{6} \sqrt{108}=\sqrt{3}$

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