Question

If the vertices of a tetrahedron have the position vectors $\overrightarrow{0}$, $\overrightarrow{i}+\overrightarrow{j},$ $2\overrightarrow{j}-\overrightarrow{k}$ and $\overrightarrow{i}+\overrightarrow{k}$ then the volume of the tetrahedron is

• A. $\frac{1}{6}$
• B. $1$
• C. $2$
• D. None of these

Answer 1

Answer: (A)

$\frac{1}{6}$

Volume of the tetrahedron $=\frac{1}{6}[\overrightarrow{i}+\overrightarrow{j}2\overrightarrow{j}-\overrightarrow{k}\overrightarrow{i}+\overrightarrow{k}]$
$=\frac{1}{6}(\overrightarrow{i}+\overrightarrow{j})\cdot \left[\right(2\overrightarrow{j}-\overrightarrow{k})\times (\overrightarrow{i}+\overrightarrow{k}\left)\right]$
$=\frac{1}{6}(\overrightarrow{i}+\overrightarrow{j})\cdot [-2\overrightarrow{k}+2\overrightarrow{i}-\overrightarrow{j}]$
$=\frac{1}{6}(2-1)$
$=\frac{1}{6}$