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Home  >>  CBSE XII  >>  Math  >>  Vector Algebra
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Show that \( ( \overrightarrow a - \overrightarrow b) \) x \( ( \overrightarrow a + \overrightarrow b) \) = 2\( ( \overrightarrow a\) x \( \overrightarrow b) \)

$\begin{array}{1 1}(A) (\overrightarrow a\times\overrightarrow b)\\(B) (\overrightarrow b\times\overrightarrow b) \\(C) 2(\overrightarrow a\times\overrightarrow b) \\ (D) 4(\overrightarrow a\times\overrightarrow b)\end{array} $

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1 Answer

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Toolbox:
  • $ \overrightarrow a\times\overrightarrow a = \overrightarrow 0$
  • $ \overrightarrow a\times \overrightarrow b= -(\overrightarrow b\times \overrightarrow a)$
Step 1:
$ (\overrightarrow a-\overrightarrow b)\times(\overrightarrow a+ \overrightarrow b)= 2(\overrightarrow a\times \overrightarrow b)$
$(\overrightarrow a-\overrightarrow b)\times (\overrightarrow a+\overrightarrow b)=\overrightarrow a\times (\overrightarrow a+\overrightarrow b)-\overrightarrow b\times (\overrightarrow a+\overrightarrow b)$
Step 2:
On expanding we get
$(\overrightarrow a\times \overrightarrow a)+(\overrightarrow a\times \overrightarrow b)-(\overrightarrow b\times \overrightarrow a)-(\overrightarrow b\times \overrightarrow b)$
But we know $(\overrightarrow a\times \overrightarrow a)=(\overrightarrow b\times \overrightarrow b)=0$
Also $(\overrightarrow a\times \overrightarrow b)=-(\overrightarrow b\times \overrightarrow a)$
Step 3:
Therefore $ (\overrightarrow a-\overrightarrow b)\times(\overrightarrow a+ \overrightarrow b)= 2(\overrightarrow a\times \overrightarrow b)$
Hence proved.
answered May 22, 2013 by sreemathi.v
 

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