Question

The volume of a tetrahedron determined by the vectors $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ is $\frac{3}{4}$ cubic units. The volume (in cubic units) of a tetrahedron determined by the vectors $3\left(\overrightarrow{a}\times \overrightarrow{b}\right),4\left(\overrightarrow{b}\times \overrightarrow{c}\right)$ and $5\left(\overrightarrow{c}\times \overrightarrow{a}\right)$ will be

Answer 1

Answer: 202.5

$\frac{1}{6}\left[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}\right]=\frac{3}{4}\Rightarrow \left[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}\right]=\frac{9}{2}\dots \left(1\right)$
Volume of tetrahedron $=\frac{1}{6}\left[3\left(\overrightarrow{a}\times \overrightarrow{b}\right)4\left(\overrightarrow{b}\times \overrightarrow{c}\right)5\left(\overrightarrow{c}\times \overrightarrow{a}\right)\right]$
$=\frac{1}{6}\times 3\times 4\times 5{\left[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}\right]}^2$
$=10\times \frac{81}{4}=202.5$ cubic units