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Home  >>  CBSE XII  >>  Math  >>  Vector Algebra
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Given that \( \overrightarrow a . \overrightarrow b = 0\) and \( \overrightarrow a\) x \(\overrightarrow b = \overrightarrow 0\) What can you conclude about the vectors \( \overrightarrow a\) and \( \overrightarrow b\)?

$\begin{array}{1 1} (A) \overrightarrow a\:and\:\overrightarrow b \;are\; parallel \\ (B) \overrightarrow a\:and\:\overrightarrow b \;are\; \perp \\ (C) |\overrightarrow a|=0\:or\:|\overrightarrow b|= 0 \\ (D) |\overrightarrow a|=|\overrightarrow b|\end{array} $

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  • $\overrightarrow a.\overrightarrow b=0$ if $\overrightarrow a\perp \overrightarrow b$
  • $\overrightarrow a\times \overrightarrow b=0$ if $\overrightarrow a$ is parallel to $\overrightarrow b$
Step 1:
Given $\overrightarrow a.\overrightarrow b=0$ and $\overrightarrow a\times \overrightarrow b=0$
$\overrightarrow a.\overrightarrow b=0$ and also $\overrightarrow a\times \overrightarrow b=0$ implies that either $|\overrightarrow a|=0$ or $|\overrightarrow b|=0$
Step 2:
This is because $\overrightarrow a.\overrightarrow b=0$ if $\overrightarrow a\perp \overrightarrow b$ and $\overrightarrow a\times \overrightarrow b=0$ if $\overrightarrow a$ is parallel to $\overrightarrow b$
Two vectors cannot be both parallel and perpendicular.So either $|\overrightarrow a|=0$ or $|\overrightarrow b|=0$
answered May 22, 2013 by sreemathi.v

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