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  4. The volume (in cubic units) of the tetrahedron with edges hat i + widehat j + hat k , hat i - hat j + hat k and hat i +2 hat j - hat k is

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Q. The volume (in cubic units) of the tetrahedron with edges $\hat{ i }+\widehat{ j }+\hat{ k }, \hat{ i }-\hat{ j }+\hat{ k }$ and $\hat{ i }+2 \hat{ j }-\hat{ k }$ is

EAMCETEAMCET 2007

Solution:

We know that, volume of tetrahedron
$=\frac{1}{6}[\overrightarrow{ A B } \overrightarrow{ A C } \overrightarrow{ A D }]$
$=\frac{1}{6}\begin{vmatrix} 1 & 1 & 1 \\ 1 & -1 & 1 \\ 1 & 2 & -1 \end{vmatrix}$
$=\frac{1}{6}[1(1-2)-1(-1-1)+1(2+1)]$
$=\frac{1}{6}[-1+2+3]$
$=\frac{4}{6}=\frac{2}{3}$