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If the position vectors of the vertices $A, B$ and $C$ are 6i, $6 \mathbf{j}$ and $\mathbf{k}$ respectively w.r.t. origin $O$, then the volume of the tetrahedron $O A B C$ is
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Verified Answer
The correct answer is:
6
Given that, the position vectors of the vertices $A, B$ and $C$ are
$$
\begin{array}{l}
\mathbf{O A}=6 \mathbf{i}=6 \mathbf{i}+0 \mathbf{j}+0 \mathbf{k} \\
\mathbf{O B}=6 \mathbf{j}=0 \mathbf{i}+6 \mathbf{j}+0 \mathbf{k} \\
\mathbf{O C}=\mathbf{k}=0 \mathbf{i}+0 \mathbf{j}+\mathbf{k}
\end{array}
$$
Now, volume of the tetrahedron
$$
\begin{array}{l}
=\frac{1}{6}[\mathbf{O A} \mathbf{O B} \mathbf{O C}] \\
=\frac{1}{6}\left|\begin{array}{lll}
6 & 0 & 0 \\
0 & 6 & 0 \\
0 & 0 & 1
\end{array}\right| \\
=\frac{1}{6}(6 \times 6 \times 1)=6
\end{array}
$$
$$
\begin{array}{l}
\mathbf{O A}=6 \mathbf{i}=6 \mathbf{i}+0 \mathbf{j}+0 \mathbf{k} \\
\mathbf{O B}=6 \mathbf{j}=0 \mathbf{i}+6 \mathbf{j}+0 \mathbf{k} \\
\mathbf{O C}=\mathbf{k}=0 \mathbf{i}+0 \mathbf{j}+\mathbf{k}
\end{array}
$$
Now, volume of the tetrahedron
$$
\begin{array}{l}
=\frac{1}{6}[\mathbf{O A} \mathbf{O B} \mathbf{O C}] \\
=\frac{1}{6}\left|\begin{array}{lll}
6 & 0 & 0 \\
0 & 6 & 0 \\
0 & 0 & 1
\end{array}\right| \\
=\frac{1}{6}(6 \times 6 \times 1)=6
\end{array}
$$
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