Q78 / 160 · TS EAMCET 2007 · Mathematics

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

  • A
    $4$
  • Correct answer
    B
    $2 / 3$
  • C
    $1 / 6$
  • D
    $1 / 3$

Solution

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

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

Relevant Questions