[ ( a + b ) ( b + c ) ( c + a ) ] =

Question

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

Smart Achievers · Accepted Answer

(3)

Ans. $2 [\vec{a} \vec{b} \vec{c}]$

Explanation:

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

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

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

$= \vec{a} \cdot (\vec{b} \times \vec{c}) + \vec{b} \cdot (\vec{c} \times \vec{a})$ $[\begin{matrix} ∵ \vec{a} \cdot (\vec{b} \times \vec{a}) = \vec{a} \cdot (\vec{c} \times \vec{a}) = 0 \\ \vec{b} \cdot (\vec{b} \times \vec{c}) = \vec{b} \cdot (\vec{b} \times \vec{a}) = 0 \end{matrix}]$

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

Solution

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

1 $0$

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

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

4 None of these

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