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If $\hat{a}$ and $\hat{b}$ are two unit vectors, then the vector
$(\hat{a}+\hat{b}) \times(\hat{a} \times \hat{b})$ is parallel to
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$(\hat{a}+\hat{b}) \times(\hat{a} \times \hat{b})$ is parallel to
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Verified Answer
The correct answer is:
$\quad(\hat{a}-\hat{b})$
$\begin{aligned} & \\ \text { } &(\hat{a}+\hat{b}) \times(\hat{a} \times \hat{b}) \\ &=\hat{a} \times(\hat{a} \times \hat{b})+\hat{b} \times(\hat{a} \times \hat{b}) \\ &=(\hat{a} \cdot \hat{b}) \hat{a}-(\hat{a} \cdot \hat{a}) \hat{b}+(\hat{b} \cdot \hat{b}) \hat{a}-(\hat{b} \cdot \hat{a}) \hat{b} \\ &=\lambda \hat{a}-\hat{b}+\hat{a}-\lambda \hat{b} \\ &=\lambda(\hat{a}-\hat{b})+1(\hat{a}-\hat{b}) . \\ &=(\lambda+1)(\hat{a}-\hat{b}) \end{aligned}$
So, it is parallel to $\hat{a}-\hat{b}$.
So, it is parallel to $\hat{a}-\hat{b}$.
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