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Questions  >>  JEEMAIN and NEET  >>  Mathematics  >>  Class12  >>  Vector Algebra
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Q)

If $\overrightarrow a,\:\overrightarrow b\:and\:\overrightarrow c$ are non coplanar vectors and if $\overrightarrow a'=\large\frac{\overrightarrow b\times \overrightarrow c}{[\overrightarrow a\:\overrightarrow b\:\overrightarrow c]},\:\overrightarrow b'=\large\frac{\overrightarrow c\times \overrightarrow a}{[\overrightarrow a\:\overrightarrow b\:\overrightarrow c]},\:and \:\overrightarrow c'=\large\frac{\overrightarrow a\times \overrightarrow b}{[\overrightarrow a\:\overrightarrow b\:\overrightarrow c]},$ then $\overrightarrow a.\overrightarrow a'+\overrightarrow b.\overrightarrow b'+\overrightarrow c.\overrightarrow c'=?$

$\begin{array}{1 1} 0 \\3[\overrightarrow a\:\overrightarrow b\:\overrightarrow c] \\ 3 \\ [\overrightarrow a\:\overrightarrow b\:\overrightarrow c]\end{array} $

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A)
$\overrightarrow a.\overrightarrow a'=\overrightarrow a.\large\frac{\overrightarrow a.(\overrightarrow b\times\overrightarrow c)}{[\overrightarrow a\:\overrightarrow b\:\overrightarrow c]}=\overrightarrow a.\large\frac{[\overrightarrow a\:\overrightarrow b\:\overrightarrow c]}{[\overrightarrow a\:\overrightarrow b\:\overrightarrow c]}=1$
similarly we can prove $\overrightarrow b.\overrightarrow b'=\overrightarrow c.\overrightarrow c'=1$
$\therefore\:\overrightarrow a.\overrightarrow a'+\overrightarrow b.\overrightarrow b'+\overrightarrow c.\overrightarrow c'=3$
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