Q. If $\overrightarrow{\mathbf{a}}\mathbf{=}2\widehat{\mathbf{i}}\mathbf{-}3\widehat{\mathbf{j}}\mathbf{+}5\widehat{\mathbf{k}}$, $\overrightarrow{\mathbf{b}}\mathbf{=}3\widehat{\mathbf{i}}\mathbf{-}4\widehat{\mathbf{j}}\mathbf{+}5\widehat{\mathbf{k}}$ and $\overrightarrow{\mathbf{c}}\mathbf{=}5\widehat{\mathbf{i}}\mathbf{-}3\widehat{\mathbf{j}}\mathbf{-}2\widehat{\mathbf{k}}$, then the volume of the parallelopiped with coterminous edges $\overrightarrow{\mathbf{a}}\mathbf{+}\overrightarrow{\mathbf{b}}\mathbf{,}\overrightarrow{\mathbf{b}}\mathbf{+}\overrightarrow{\mathbf{c}}\mathbf{,}\overrightarrow{\mathbf{c}}\mathbf{+}\overrightarrow{\mathbf{a}}$ is
Explanation
(d)
Given that, $\overrightarrow{\mathbf{a}} = 2\widehat{\mathbf{i}} - 3\widehat{\mathbf{j}} + 5\widehat{\mathbf{k}}$, $\overrightarrow{\mathbf{b}} = 3\widehat{\mathbf{i}} - 4\widehat{\mathbf{j}} + 5\widehat{\mathbf{k}}$ and $\overrightarrow{\mathbf{c}} = 5\widehat{\mathbf{i}} - 3\widehat{\mathbf{j}} - 2\widehat{\mathbf{k}}$
$\therefore$ Volume of parallelopiped where sides are $\overrightarrow{\mathbf{a}}\mathbf{+}\overrightarrow{\mathbf{b}}\mathbf{,\ }\overrightarrow{\mathbf{b}}\mathbf{+}\overrightarrow{\mathbf{c}}$ and $\overrightarrow{\mathbf{c}}\mathbf{+}\overrightarrow{\mathbf{a}}$, is
$\left\lbrack \ \overrightarrow{\mathbf{a}}\mathbf{\ }\overrightarrow{\mathbf{b}}\mathbf{\ }\overrightarrow{\mathbf{c}}\ \right\rbrack = \left| \begin{matrix}
2 & - 3 & 5 \\
3 & - 4 & 5 \\
5 & - 3 & - 2
\end{matrix} \right|$
$= \left\lbrack 2(8 + 15) + 3( - 6 - 25) + 5( - 9 + 20) \right\rbrack$
$= 46 - 93 + 55 = 8$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.