Q62 / 180 · COMEDK 2018 · Mathematics

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

  • Correct answer
    A
    $\frac{5}{6}$
  • B
    $\frac{6}{5}$
  • C
    5
  • D
    $\frac{2}{3}$

Solution

Let $A(1,1,1), B(2,1,3), C(3,2,2)$ and $D(3,3,4)$. So, $\mathbf{A B}=\hat{\mathbf{i}}+2 \hat{\mathbf{k}}, \mathbf{A C}=2 \hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}$ and $\mathbf{A D}=2 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}$ $$ \begin{aligned} &\text { Volume of tetrahedron }=\frac{1}{6}\left[\begin{array}{lll} 1 & 0 & 2 \\ 2 & 1 & 1 \\ 2 & 2 & 3 \end{array}\right] \\ &\quad=\frac{1}{6}|1(3-2)+0(6-2)+2(4-2)| \\ &=\frac{1}{6}(1+0+4)=\frac{5}{6} \text { cubic units } \end{aligned} $$

Hence, the correct answer is option: 'A'.

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