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  4. The scalar A ⋅[( B + C ) ×( A + B + C )] equals

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Q. The scalar $\overrightarrow{ A } \cdot[(\overrightarrow{ B }+\overrightarrow{ C }) \times(\overrightarrow{ A }+\overrightarrow{ B }+\overrightarrow{ C })]$ equals

IIT JEEIIT JEE 1982Vector Algebra

Solution:

$\overrightarrow{ A } \cdot\{(\overrightarrow{ B }+\overrightarrow{ C }) \times(\overrightarrow{ A }+\overrightarrow{ B }+\overrightarrow{ C })\}$
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$[\because$ it is a scalar triple product of three vectors of the form $\overrightarrow{ A }, \overrightarrow{ B }+\overrightarrow{ C }, \overrightarrow{ A }+\overrightarrow{ B }+\overrightarrow{ C }]$
$= \overrightarrow{ A } \cdot(\overrightarrow{ B } \times \overrightarrow{ A }+\overrightarrow{ B } \times \overrightarrow{ B }$
$+\overrightarrow{ B } \times \overrightarrow{ C }+\overrightarrow{ C } \times \overrightarrow{ A }+\overrightarrow{ C } \times \overrightarrow{ B }+\overrightarrow{ C } \times \overrightarrow{ C }) $
$=\overrightarrow{ A } \cdot(\overrightarrow{ B } \times \overrightarrow{ A })+\overrightarrow{ A } \cdot(\overrightarrow{ B } \times \overrightarrow{ C })+\overrightarrow{ A } \cdot(\overrightarrow{ C } \times \overrightarrow{ B })$
$=[\overrightarrow{ A } \overrightarrow{ B } \overrightarrow{ A }]-[\overrightarrow{ A } \overrightarrow{ B } \overrightarrow{ C }]=0$