( a + b ). ( b + c ) ( a + b + c )= 1. 0 2. - [ a b c ] 3. 2 [ a b c ] 4. [ a b c ]

4. [ a b c ] Solution: We have ( a + b ). ( b + c ) ( a + b + c ) = ( a + b . [ ( b + c ) a + ( b + c ) b + ( b + c ) c ] = ( a + b ). ( b a + c a + b b + c b + b c + c c ) = ( a + b ). ( b a + c a +0+ c b + b c +0 = ( a + b ). ( b a + c a - b c + b c ) = ( a + b ). ( b a + c a ) = a ( b a )+ b . ( b a )+ a . ( c a )+ b . ( c a ) =0+0+0+ [ b c a ] = [ b c a ] = [ a b c ]

( a + b ). ( b + c ) ( a + b + c )= 1. 0 2. - [ a b c ] 3. 2 [ a …

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Question

$\big(\vec{\text{a}}+\vec{\text{b}}\big).\big(\vec{\text{b}}+\vec{\text{c}}\big)\times\big(\vec{\text{a}}+\vec{\text{b}}+\vec{\text{c}}\big)=$

$0$
$-\big[\vec{\text{a}}\vec{\text{b}}\vec{\text{c}}\big]$
$2\big[\vec{\text{a}}\vec{\text{b}}\vec{\text{c}}\big]$
$\big[\vec{\text{a}}\vec{\text{b}}\vec{\text{c}}\big]$

✓

Answer

$\big[\vec{\text{a}}\vec{\text{b}}\vec{\text{c}}\big]$

Solution:

We have

$\big(\vec{\text{a}}+\vec{\text{b}}\big).\big(\vec{\text{b}}+\vec{\text{c}}\big)\times\big(\vec{\text{a}}+\vec{\text{b}}+\vec{\text{c}}\big)$

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

$=\big(\vec{\text{a}}+\vec{\text{b}}\big).\big(\vec{\text{b}}\times\vec{\text{a}}+\vec{\text{c}}\times\vec{\text{a}}+\vec{\text{b}}\times\vec{\text{b}}+\vec{\text{c}}\times\vec{\text{b}}+\vec{\text{b}}\times\vec{\text{c}}+\vec{\text{c}}\times\vec{\text{c}}\big)$

$=\big(\vec{\text{a}}+\vec{\text{b}}\big).\big(\vec{\text{b}}\times\vec{\text{a}}+\vec{\text{c}}\times\vec{\text{a}}+0+\vec{\text{c}}\times\vec{\text{b}}+\vec{\text{b}}\times\vec{\text{c}}+0$

$=\big(\vec{\text{a}}+\vec{\text{b}}\big).\big(\vec{\text{b}}\times\vec{\text{a}}+\vec{\text{c}}\times\vec{\text{a}}-\vec{\text{b}}\times\vec{\text{c}}+\vec{\text{b}}\times\vec{\text{c}}\big)$

$=\big(\vec{\text{a}}+\vec{\text{b}}\big).\big(\vec{\text{b}}\times\vec{\text{a}}+\vec{\text{c}}\times\vec{\text{a}}\big)$

$=\vec{\text{a}}\big(\vec{\text{b}}\times\vec{\text{a}}\big)+\vec{\text{b}}.\big(\vec{\text{b}}\times\vec{\text{a}}\big)+\vec{\text{a}}.\big(\vec{\text{c}}\times\vec{\text{a}}\big)+\vec{\text{b}}.\big(\vec{\text{c}}\times\vec{\text{a}}\big)$

$=0+0+0+\big[\vec{\text{b}}\vec{\text{c}}\vec{\text{a}}\big]$

$=\big[\vec{\text{b}}\vec{\text{c}}\vec{\text{a}}\big]$

$=\big[\vec{\text{a}}\vec{\text{b}}\vec{\text{c}}\big]$

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