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If the volume of the parallelopiped with $\overrightarrow{\mathbf{a}}, \overrightarrow{\mathbf{b}}$ and $\overrightarrow{\mathbf{c}}$ as coterminous edges is $40 \mathrm{cu}$ unit, then the volume of the parallelopiped having $\overrightarrow{\mathbf{b}}+\overrightarrow{\mathbf{c}}, \overrightarrow{\mathbf{c}}+\overrightarrow{\mathbf{a}}$ and $\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}}$ as coterminous edges in cubic unit is
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Verified Answer
The correct answer is:
80
Given, volume of parallelopiped
$$
[\overrightarrow{\mathbf{a}} \overrightarrow{\mathbf{b}} \overrightarrow{\mathbf{c}}]=40
$$
$\therefore$ Volume of parallelopiped
$=\left[\begin{array}{lll}\overrightarrow{\mathbf{b}}+\overrightarrow{\mathbf{c}} & \overrightarrow{\mathbf{c}}+\overrightarrow{\mathbf{a}} & \overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}}\end{array}\right]$
$=2[\overrightarrow{\mathbf{a}} \overrightarrow{\mathbf{b}} \overrightarrow{\mathbf{c}}]$
$=2 \times 40=80 \mathrm{cu}$ unit
$$
[\overrightarrow{\mathbf{a}} \overrightarrow{\mathbf{b}} \overrightarrow{\mathbf{c}}]=40
$$
$\therefore$ Volume of parallelopiped
$=\left[\begin{array}{lll}\overrightarrow{\mathbf{b}}+\overrightarrow{\mathbf{c}} & \overrightarrow{\mathbf{c}}+\overrightarrow{\mathbf{a}} & \overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}}\end{array}\right]$
$=2[\overrightarrow{\mathbf{a}} \overrightarrow{\mathbf{b}} \overrightarrow{\mathbf{c}}]$
$=2 \times 40=80 \mathrm{cu}$ unit
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