Let $\hat{a} and \hat{b}$ be mutually perpendicular unit vectors. Then for any arbitrary $\vec{r}$

Moderate

A

$\vec{r} = (\vec{r} \cdot \hat{a}) \hat{a} + (\vec{r} \cdot \hat{b}) \hat{b} + (\vec{r} \cdot (\hat{a} \times \hat{b})) (\hat{a} \times \hat{b})$
B

$\vec{r} = (\vec{r} \cdot \hat{a}) - (\vec{r} \cdot \hat{b}) \hat{b} - (\vec{r} \cdot (\hat{a} \times \hat{b})) (\hat{a} \times \hat{b})$
C

$\vec{r} = (\vec{r} \cdot \hat{a}) \hat{a} - (\vec{r} \cdot \hat{b}) \hat{b} + (\vec{r} \cdot (\hat{a} \times \hat{b})) (\hat{a} \times \hat{b})$
D

none of these

Solution

$\begin{array}{l} Let \vec{r} = x_{1} \hat{a} + x_{2} \hat{b} + x_{3} (\hat{a} \times \hat{b}) \\ \Rightarrow \vec{r} \cdot \hat{a} = x_{1} + x_{2} \hat{a} \cdot \hat{b} + x_{3} \hat{a} \cdot (\hat{a} \times \hat{b}) = x_{1} \\ Also, \vec{r} \cdot \hat{b} = x_{1} \hat{a} \cdot \hat{b} + x_{2} + x_{3} \hat{b} \cdot (\hat{a} + \hat{b}) = x_{2} \\ and \vec{r} \cdot (\hat{a} \times \hat{b}) = x_{1} \hat{a} \cdot (\hat{a} \times \hat{b}) + x_{2} \hat{b} \cdot (\hat{a} \times \hat{b}) + x_{3} (\hat{a} \times \hat{b}) \cdot (\hat{a} \times \hat{b}) = x_{3} \\ \Rightarrow \vec{r} = (\vec{r} \cdot \hat{a}) \hat{a} + (\vec{r} \cdot \hat{b}) \hat{b} + (\vec{r} \cdot \hat{a} \times \hat{b})) (\hat{a} \times \hat{b}) \end{array}$

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