Question

If the volume of a parallelepiped with $\overrightarrow{a}\times \overrightarrow{b},\overrightarrow{b}\times \overrightarrow{c},\overrightarrow{c}\times \overrightarrow{a}$ as co-termmus edges is $9cu$ units, then the volume of the parallelepiped with $(a\times b)\times (b\times c),(b\times c)\times (c\times a),$ $(c\times a)\times (a\times b)$ as co-terminus edges is

• A. 9 cu unit
• B. 729 cu unit
• C. 81 cu unit
• D. 27 cu unit
• E. 243 cu unit

Answer 1

Answer: (C)

Let $\overrightarrow{A}=\overrightarrow{a}\times \overrightarrow{b},\overrightarrow{B}=\overrightarrow{b}\times \overrightarrow{c},\overrightarrow{C}=\overrightarrow{c}\times \overrightarrow{a}$
Given, $[\overrightarrow{A}\overrightarrow{B}\overrightarrow{C}]=9cuunit$ Now, $[\overrightarrow{A}\times \overrightarrow{B}\overrightarrow{B}\times \overrightarrow{C}\overrightarrow{C}\times \overrightarrow{A}]$
$=(\overrightarrow{A}\times \overrightarrow{B}).[(\overrightarrow{B}\times \overrightarrow{C})\times (\overrightarrow{C}\times \overrightarrow{A})]$
$=(\overrightarrow{A}\times \overrightarrow{B}).[[(\overrightarrow{B}\times \overrightarrow{C}).\overrightarrow{A}]\overrightarrow{C}-[(\overrightarrow{B}\times \overrightarrow{C}).\overrightarrow{C}]\overrightarrow{A}]$
$=(\overrightarrow{A}\times \overrightarrow{B}).[[(\overrightarrow{B}\overrightarrow{C}\overrightarrow{A}]\overrightarrow{C}-0]$
$={[\overrightarrow{A}\overrightarrow{B}\overrightarrow{C}]}^2=9^2$
$=81cuunit$