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

$|\overrightarrow a\times\overrightarrow b|^2= ?$ (which one of the following )

$\begin{array}{1 1} (i)\:\:\left |\begin {array}{cc} \overrightarrow a.\overrightarrow a & \overrightarrow a.\overrightarrow a \\ \overrightarrow a.\overrightarrow b & \overrightarrow b.\overrightarrow b\end {array}\right| \:\:\: & (ii)\:\:\:\left |\begin {array}{cc} \overrightarrow a.\overrightarrow a & -\overrightarrow a.\overrightarrow a \\ \overrightarrow a.\overrightarrow b & \overrightarrow b.\overrightarrow b\end {array}\right| \:\:\:\\ (iii)\:\: \:\left |\begin {array}{cc} \overrightarrow a.\overrightarrow a & \overrightarrow a.\overrightarrow a \\ -\overrightarrow a.\overrightarrow b & \overrightarrow b.\overrightarrow b\end {array}\right| \:\:\: & (iv)\:\:\:\left |\begin {array}{cc} \overrightarrow a.\overrightarrow a & -\overrightarrow a.\overrightarrow a \\ \overrightarrow a.\overrightarrow b &- \overrightarrow b.\overrightarrow b\end {array}\right| \end{array} $

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A)
$|\overrightarrow a\times \overrightarrow b|^2=|\overrightarrow a|^2\:|\overrightarrow b|^2.sin^2\theta$
where $\theta$ is the angle between $\overrightarrow a\:\:and\:\:\overrightarrow b$
$\left |\begin {array} {cc} \overrightarrow a.\overrightarrow a & \overrightarrow a.\overrightarrow b\\ \overrightarrow a.\overrightarrow b & \overrightarrow b.\overrightarrow b\end {array}\right |=|\overrightarrow a|^2\:|\overrightarrow b|^2-(\overrightarrow a.\overrightarrow b)^2$
$=|\overrightarrow a|^2\:|\overrightarrow b|^2-|\overrightarrow a|^2 |\overrightarrow b|^2cos^2\theta$
$=|\overrightarrow a|^2\:|\overrightarrow b|^2\:(1-cos^2\theta)$
$=|\overrightarrow a|^2\:|\overrightarrow b|^2\:sin^2\theta=|\overrightarrow a\times\overrightarrow b|^2$
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