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Questions  >>  CBSE XII  >>  Math  >>  Vector Algebra
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Q)

The value of the expression ${\mid \overrightarrow{a}\times \overrightarrow{b}\mid}^2+(\overrightarrow{a}.\overrightarrow{b})^2$ is _________.

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A)
Toolbox:
  • $ |\overrightarrow a\times \overrightarrow b |=|\overrightarrow a|.|\overrightarrow b| \: sin\theta \;\hat n$
  • $ |\overrightarrow a\times \overrightarrow b |=|\overrightarrow a|.|\overrightarrow b| \: sin\theta$
$ |\overrightarrow a\times \overrightarrow b |^2+(a.b)^2=|\overrightarrow a|^2.|\overrightarrow b|^2 $
$ |\overrightarrow a\times \overrightarrow b |=|\overrightarrow a|.|\overrightarrow b| \: sin\theta$
On squaring we get,
$ |\overrightarrow a\times \overrightarrow b |^2=|\overrightarrow a|^2.|\overrightarrow b|^2 \: sin^2\theta$
But $ \sin ^2 \theta=(1-\cos \theta)$
$ |\overrightarrow a\times \overrightarrow b |^2=|\overrightarrow a|^2.|\overrightarrow b|^2 -|\overrightarrow a|^2 |\overrightarrow b|^2 \cos ^2 \theta$
$ \qquad \qquad=|\overrightarrow a|^2.|\overrightarrow b|^2 -(|\overrightarrow a| |\overrightarrow b| \cos \theta)^2$
But $\overrightarrow a.\overrightarrow b=(| \;a\;| |\;b\;| \cos \theta)$
$ |\overrightarrow a\times \overrightarrow b |^2=|\overrightarrow a|^2.|\overrightarrow b|^2-(\overrightarrow a.\overrightarrow b)^2 $
=>$ |\overrightarrow a\times \overrightarrow b |^2+(\overrightarrow a.\overrightarrow b)^2 =|\overrightarrow a|^2.|\overrightarrow b|^2$
The value of the expression ${\mid \overrightarrow{a}\times \overrightarrow{b}\mid}^2+(\overrightarrow{a}.\overrightarrow{b})^2$ is $ |\overrightarrow a|^2 |\overrightarrow b|^2$
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