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  4. If the vectors a,b and c from the sides BC, CA and AB respectively, of a triangle ABC, then

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Q. If the vectors $\overrightarrow{a},\overrightarrow{b}$ and $\overrightarrow{c}$ from the sides $BC, CA$ and $AB$ respectively, of a triangle $ABC$, then

IIT JEEIIT JEE 2000Vector Algebra

Solution:

By triangle law, $\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}=\overrightarrow{0}$
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Taking cross product by $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ respectively,
$\overrightarrow{a}\times(\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c})=\overrightarrow{a}\times\overrightarrow{0}=\overrightarrow{0}$
$\Rightarrow \overrightarrow{a}\times\overrightarrow{a}+\overrightarrow{a}\times\overrightarrow{b}+\overrightarrow{a}\times\overrightarrow{c}=\overrightarrow{a}$
$\Rightarrow \overrightarrow{a}\times\overrightarrow{b}=\overrightarrow{c}\times\overrightarrow{a} [\because \overrightarrow{a}\times\overrightarrow{a}=\overrightarrow{o}]$
Similarly, $\overrightarrow{a}\times\overrightarrow{b}=\overrightarrow{b}\times\overrightarrow{a}$
$\therefore \overrightarrow{a}\times\overrightarrow{b}=\overrightarrow{b}\times\overrightarrow{c}\times\overrightarrow{a}$