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Let $u$ and $v$ be two non-zero vectors in $R^3$. Then $|u \times v|^2+|u \cdot v|^2$ is equal to
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Verified Answer
The correct answer is:
$|u|^2|v|^2$
Given $u$ and $v$ are non zero vectors in $I R^3$.
$\begin{aligned} & \text { To Find }|u \times v|^2+|u \cdot v|^2 \\ & \qquad \begin{array}{l}|u|^2|v|^2 \sin ^2 \theta+|u|^2|v|^2 \cos ^2 \theta \\ =|u|^2|v|^2\left(\sin ^2 \theta+\cos ^2 \theta\right) \\ \quad\{\because u \times v=|u||v| \sin \theta u \cdot v=|u||v| \cos \theta\}\end{array}\end{aligned}$
$=|u|^2|v|^2$
$\begin{aligned} & \text { To Find }|u \times v|^2+|u \cdot v|^2 \\ & \qquad \begin{array}{l}|u|^2|v|^2 \sin ^2 \theta+|u|^2|v|^2 \cos ^2 \theta \\ =|u|^2|v|^2\left(\sin ^2 \theta+\cos ^2 \theta\right) \\ \quad\{\because u \times v=|u||v| \sin \theta u \cdot v=|u||v| \cos \theta\}\end{array}\end{aligned}$
$=|u|^2|v|^2$
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