Question

If three non-coplanar vectors $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ from a parallelopiped of volume 8 cubic units, then the values of $\left[3\overrightarrow{a}4\overrightarrow{b}5\overrightarrow{c}\right]$ are _________________.

Answer 1

Volume of parallelopiped is 8 cubic units $\left[3\overrightarrow{a}4\overrightarrow{b}5\overrightarrow{c}\right]$
$$\begin{gathered} =\pm 3\times 4\times 5\left[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}\right] \\ =\pm 60\left[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}\right] \\ =\pm 60\times 8 \\ =\pm 480 \\ \mathrm{i}.\mathrm{e}\left[3\overrightarrow{a}4\overrightarrow{b}5\overrightarrow{c}\right]=\pm 480 \end{gathered}$$