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  4. If ab and c are three non-coplanar vectors, then (a+b+c).[(a+b)×(a+c)] equals

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Q. If $\overrightarrow{a}\overrightarrow{b}$ and $\overrightarrow{c}$ are three non-coplanar vectors, then $(\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}).[(\overrightarrow{a}+\overrightarrow{b})\times(\overrightarrow{a}+\overrightarrow{c})]$ equals

IIT JEEIIT JEE 1995Vector Algebra

Solution:

$(\overrightarrow{ a }+\overrightarrow{ b }+\overrightarrow{ c }) \cdot[(\overrightarrow{ a }+\overrightarrow{ b }) \times(\overrightarrow{ a }+\overrightarrow{ c })] $
$=(\overrightarrow{ a }+\overrightarrow{ b }+\overrightarrow{ c }) \cdot[\overrightarrow{ a } \times \overrightarrow{ a }+\overrightarrow{ a } \times \overrightarrow{ c }+\overrightarrow{ b } \times \overrightarrow{ a }+\overrightarrow{ b } \times \overrightarrow{ c }]$
$=\{\overrightarrow{ a } \cdot(\overrightarrow{ a } \times \overrightarrow{ c })+\overrightarrow{ a } \cdot (\overrightarrow{ b } \times \overrightarrow{ a })+\overrightarrow{ a } \cdot(\overrightarrow{ b } \times \overrightarrow{ c })\}+\{\overrightarrow{ b } \cdot(\mid \overrightarrow{ a } \times \overrightarrow{ c })$
$+\overrightarrow{ b } \cdot(\overrightarrow{ b } \times \overrightarrow{ a })+\overrightarrow{ b } \cdot(\overrightarrow{ b } \times \overrightarrow{ c })\}+\{\overrightarrow{ c } \cdot(\overrightarrow{ a } \times \overrightarrow{ c })+\overrightarrow{ c } \cdot(\overrightarrow{ b } \times \overrightarrow{ a })+\overrightarrow{ c } \cdot(\overrightarrow{ b } \times \overrightarrow{ c })\}$
$\Rightarrow {[\overrightarrow{ a } \overrightarrow{ b } \overrightarrow{ c }]+[\overrightarrow{ b } \overrightarrow{ a } \overrightarrow{ c }]+[\overrightarrow{ c } \overrightarrow{ b } \overrightarrow{ a }] \Rightarrow[\overrightarrow{ a } \overrightarrow{ b } \overrightarrow{ c }] }$