( a b ) [ ( b c ) ( a b + b c + c a ) ] is - Vidyadip

MCQ

$\left( {\vec a \times \vec b} \right) \times \left[ {\left( {\vec b \times \vec c} \right) \times \left( {\vec a \times \vec b + \vec b \times \vec c + \vec c \times \vec a} \right)} \right]$ is

A
$\left[ {\vec a\,\vec b\,\vec c} \right]\left[ {\left( {\vec b \cdot \vec a + \vec a \cdot \vec c} \right)\vec b - \left( {{{\left| {\vec b} \right|}^2} + \vec b \cdot \vec c} \right)\vec a} \right]$
B
$\left[ {\vec a\,\vec b\,\vec c} \right]\left[ {\left( {\vec b \cdot \vec a + \vec a \cdot \vec c} \right)\vec b + \left( {{{\left| {\vec b} \right|}^2} - \vec b \cdot \vec c} \right)\vec a} \right]$
C
$\left[ {\vec a\,\vec b\,\vec c} \right]\left[ {\left( {\vec b \cdot \vec a - \vec a \cdot \vec c} \right)\vec b + \left( {{{\left| {\vec b} \right|}^2} + \vec b \cdot \vec c} \right)\vec a} \right]$
✓
$\left[ {\vec a\,\vec b\,\vec c} \right]\left[ {\left( {\vec a \cdot \vec c - \vec b \cdot \vec a} \right)\vec b + \left( {{{\left| {\vec b} \right|}^2} - \vec b \cdot \vec c} \right)\vec a} \right]$

✓

Answer

Correct option: D.

$\left[ {\vec a\,\vec b\,\vec c} \right]\left[ {\left( {\vec a \cdot \vec c - \vec b \cdot \vec a} \right)\vec b + \left( {{{\left| {\vec b} \right|}^2} - \vec b \cdot \vec c} \right)\vec a} \right]$

d
$(\vec{a} \times \vec{b}) \times(-\vec{b} \vec{c} \vec{a}] \vec{b}+[\vec{b} \vec{c} \vec{a}] \vec{c})$

$=[\vec{b} \vec{c} \vec{a}]\left(-(\vec{a} \cdot \vec{b}) \vec{b}+|\vec{b}|^{2} \vec{a}+(\vec{a} \cdot \vec{c}) \vec{b}-(\vec{b} \cdot \vec{c}) \vec{a}\right)$

$=[\vec{a} \vec{b} \vec{c}]\left[(\vec{a} \cdot \vec{c}-\vec{b} \cdot \vec{a}) \vec{b}+\left(|\vec{b}|^{2}-\vec{b} \cdot \vec{c}\right) \vec{a}\right]$

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