If $\mathrm{A}(-1,2,3), \mathrm{B}(3,-2,1), \mathrm{C}(2,1,3)$ and $\mathrm{D}(-1,-2,4)$ are the vertices of…

If $\mathrm{A}(-1,2,3), \mathrm{B}(3,-2,1), \mathrm{C}(2,1,3)$ and $\mathrm{D}(-1,-2,4)$ are the vertices of a tetrahedron, then its volume is
  1. $\frac{16}{3}$ Cu. units
  2. $\frac{13}{6}$ cu. units
  3. $\frac{16}{31}$ CU. units
  4. $\frac{31}{6}$ cu. units

Solution

We have $\overline{A B}=4 \hat{i}-4 \hat{j}-2 \hat{k}, \overline{A C}=3 \hat{i}-\hat{j}$ and $\overline{A D}=-4 \bar{j}+\bar{k}$ Volume of tetrahedron $=\frac{1}{6}[\overline{\mathrm{AB}} \overline{\mathrm{AC}} \overline{\mathrm{AD}}]$ $\begin{array}{l} =\frac{1}{6}\left|\begin{array}{ccc} 4 & -4 & -2 \\ 3 & -1 & 0 \\ 0 & -4 & 1 \end{array}\right| \\ =\frac{1}{6}[4(-1)+4(3)-2(-12)]=\frac{1}{6}(-4+12+24)=\frac{32}{6}=\frac{16}{3} \end{array}$

Asked in: MHT CET 2020 (14 Oct Shift 1)

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