The value of $[\vec{a}, \vec{b} + \vec{c}, \vec{a} + \vec{b} + \vec{c}]$ is

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The value of $[\vec{a}, \vec{b} + \vec{c}, \vec{a} + \vec{b} + \vec{c}]$ is

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0
$[\vec{a} \vec{b} \vec{c}]$
$2 [\vec{a} \vec{b} \vec{c}]$
$3 [\vec{a} \vec{b} \vec{c}]$

Answer 1

Concept:

If $\vec{a}, \vec{b}, \vec{c}$ are any non-zero vector then

$[\vec{a} \vec{b} \vec{c}] = \vec{a} . (\vec{b} \times \vec{c}) . . . . (1)$

Calculation:

Given, $[\vec{a}, \vec{b} + \vec{c}, \vec{a} + \vec{b} + \vec{c}]$

$= \vec{a} . [(\vec{b} + \vec{c}) \times (\vec{a} + \vec{b} + \vec{c})]$

$= \vec{a} . [\vec{b} \times \vec{a} + \vec{b} \times \vec{b} + \vec{b} \times \vec{c} + \vec{c} \times \vec{a} + \vec{c} \times \vec{b} + \vec{c} \times \vec{c}]$

$= \vec{a} . [\vec{b} \times \vec{a} + \vec{b} \times \vec{c} + \vec{c} \times \vec{a} + \vec{c} \times \vec{b}]$

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

$(s i n c e [\vec{a} \vec{b} \vec{a}] = 0)$

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

= 0

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The value of $[\vec{a}, \vec{b} + \vec{c}, \vec{a} + \vec{b} + \vec{c}]$ is