Question

If $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ are any three vectors in space then $(\overrightarrow{c}+\overrightarrow{b})\times (\overrightarrow{c}+\overrightarrow{a}).(\overrightarrow{c}+\overrightarrow{b}+\overrightarrow{a})$ is equal to

• A. $3\left[\begin{array}{ccc}\overrightarrow{a}& \overrightarrow{b}& \overrightarrow{c}\end{array}\right]$
• B. $0$
• C. $\left[\begin{array}{ccc}\overrightarrow{a}& \overrightarrow{b}& \overrightarrow{c}\end{array}\right]$
• D. None of these

Answer 1

Answer: (C)

$\left[\begin{array}{ccc}\overrightarrow{a}& \overrightarrow{b}& \overrightarrow{c}\end{array}\right]$

$(\overrightarrow{c}+\overrightarrow{b})\times (\overrightarrow{c}+\overrightarrow{a}).(\overrightarrow{c}+\overrightarrow{b}+\overrightarrow{a})$
$=(\overrightarrow{c}\times \overrightarrow{a}+\overrightarrow{b}\times \overrightarrow{c}+\overrightarrow{b}\times \overrightarrow{a})\cdot (\overrightarrow{c}+\overrightarrow{b}+\overrightarrow{a})$
$=\overrightarrow{c}\times \overrightarrow{a}\cdot \overrightarrow{c}+\overrightarrow{b}\times \overrightarrow{c}\cdot \overrightarrow{c}+\overrightarrow{b}\times \overrightarrow{a}\cdot \overrightarrow{c}$
$+\overrightarrow{c}\times \overrightarrow{a}\cdot \overrightarrow{b}+\overrightarrow{b}\times \overrightarrow{c}\cdot \overrightarrow{b}+\overrightarrow{b}\times \overrightarrow{a}\cdot \overrightarrow{b}$
$+\overrightarrow{c}\times \overrightarrow{a}\cdot \overrightarrow{a}+\overrightarrow{b}\times \overrightarrow{c}\cdot \overrightarrow{a}+\overrightarrow{b}\times \overrightarrow{a}\cdot \overrightarrow{a}$
$=0+0+\overrightarrow{b}\times \overrightarrow{a}\cdot \overrightarrow{c}+\overrightarrow{c}\times \overrightarrow{a}\cdot \overrightarrow{b}+0+0+0+\overrightarrow{b}\times \overrightarrow{c}\cdot \overrightarrow{a}+0$
$=\overrightarrow{b}\times \overrightarrow{a}\cdot \overrightarrow{c}+\overrightarrow{c}\times \overrightarrow{a}\cdot \overrightarrow{b}+\overrightarrow{b}\times \overrightarrow{c}\cdot \overrightarrow{a}$