Question

If $V$ is the volume of the parallelopiped having three coterminus edges as $\overrightarrow{a},\overrightarrow{b}$ and $\overrightarrow{c}$ then the volume of the parallelopiped having three coterminus edges as
$\overrightarrow{\alpha }=(\overrightarrow{a}.\overrightarrow{a})\overrightarrow{a}+(\overrightarrow{a}.\overrightarrow{b})\overrightarrow{b}+(\overrightarrow{a}.\overrightarrow{c})\overrightarrow{c}$, $\overrightarrow{\beta }=(\overrightarrow{a}.\overrightarrow{b})\overrightarrow{a}+(\overrightarrow{b}.\overrightarrow{b})\overrightarrow{b}+(\overrightarrow{b}.\overrightarrow{c})\overrightarrow{c}$, $\overrightarrow{\gamma }=(\overrightarrow{a}.\overrightarrow{c})\overrightarrow{a}+(\overrightarrow{b}.\overrightarrow{c})\overrightarrow{b}+(\overrightarrow{c}.\overrightarrow{c})\overrightarrow{c}$ is

• A. $V^3$
• B. $3V$
• C. $V^2$
• D. $2V$

Answer 1

Answer: (A)

$V^3$

Volume of parellelopiped having sides $\overline{a},\overline{b},\overline{c}$ is $(\overline{a},\overline{b},\overline{c})$

So volume of parallelopiped having sides $\overline{\alpha },\overline{\beta },\overline{\gamma }$ is ( $\overline{\alpha },\overline{\beta },\overline{\gamma })$

Substitute values

$(\overline{a}.\overline{a})\overline{a}+(\overline{a}.\overline{b})\overline{b}+(\overline{a}.\overline{c})\overline{c}$, $(\overline{a}.\overline{b})\overline{a}+(\overline{b}.\overline{b})\overline{b}+(\overline{b}.\overline{c})\overline{c}$, $(\overline{a}.\overline{c})\overline{a}+(\overline{b}.\overline{c})\overline{b}+(\overline{c}.\overline{c})\overline{c}$

Solving, we get

$(a^3,b^3,c^3)=V^3$