The value of ( a + 2 b - c ). ( a - b ) ( a - b - c ) is equal to - Vidyadip

MCQ

The value of $\left( {\vec a + 2\vec b - \vec c} \right).\left\{ {\left( {\vec a - \vec b} \right) \times \left( {\vec a - \vec b - \vec c} \right)} \right\}$ is equal to

A
$\left[ {\vec a\,\vec b\,\vec c} \right]$
B
$2\left[ {\vec a\,\vec b\,\vec c} \right]$
✓
$3\left[ {\vec a\,\vec b\,\vec c} \right]$
D
$4\left[ {\vec a\,\vec b\,\vec c} \right]$

✓

Answer

Correct option: C.

$3\left[ {\vec a\,\vec b\,\vec c} \right]$

c
$(\vec{a}+2 \vec{b}-\vec{c}) \cdot\{0-\vec{a} \times \vec{b}-\vec{a} \times \vec{c}-\vec{b} \times \vec{a}+0+\vec{b} \times \vec{c}\}$

$=(\vec{a}+2 \vec{b}-\vec{c}) \cdot\{\vec{b} \times \vec{c}-\vec{a} \times \vec{c}\}$

$=[\vec{a} \vec{b} \vec{c}]-0+0-2 \vec{b} \cdot(\vec{a} \times \vec{c})-0+0$

$=[\vec{a} \vec{b} \vec{c}]-2[\vec{b} \vec{a} \vec{c}]$

$=[\vec{a} \vec{b} \vec{c}]+2[\vec{a} \vec{b} \vec{c}]=3[\vec{a} \vec{b} \vec{c}]$

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