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  4. The volume of the tetrahedron formed by the points (1, 1, 1 ), (2, 1, 3), (3, 2, 2) and (3, 3, 4) in cubic units is

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Q. The volume of the tetrahedron formed by the points $(1, 1, 1 ), (2, 1, 3), (3, 2, 2)$ and $(3, 3, 4)$ in cubic units is

COMEDKCOMEDK 2009Vector Algebra

Solution:

Let $A(1, 1, 1), B(2, 1, 3), C(3, 2, 2)$ and $D(3,3, 4)$
$\bar{AB} = (2-1, 1-1, 3 -1) = (1, 0, 2)$
$\bar{AC} =(3-1, 2-1, 2-1) =(2,1,1)$
$\bar{AD} = (3-1, 3 -1, 4 -1) = (2, 2, 3)$
$V = \frac{1}{6} |[\bar{u}, \bar{v}, \bar{w}]| $
$[\bar{u}, \bar{v}, \bar{w}]\begin{vmatrix}1&0&2\\ 2&1&1\\ 2&2&3\end{vmatrix}=1\left(3-2\right)+2\left(4-2\right) $
$= 1+2\left(2\right)=5 \Rightarrow V =\frac{1}{6}.5=\frac{5}{6}$ cubic units