Vector component of a¯⊥larb¯

a¯−(a¯⋅b¯)|b¯|2⋅b¯=(b¯⋅b¯)a¯−(a¯⋅b¯)b¯|b¯|2=b¯×(a¯×b¯)|b¯|2

Vector component of a¯⊥larb¯

a¯−(a¯⋅b¯)|b¯|2⋅b¯=(b¯⋅b¯)a¯−(a¯⋅b¯)b¯|b¯|2=b¯×(a¯×b¯)|b¯|2

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Vector component of $\bar{a} ⊥ lar \bar{b}$

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$(\bar{b} . \bar{c}) \bar{a}$

$\frac{\bar{b} \times (\bar{a} \times \bar{b})}{{|\bar{b}|}^{2}}$

$\frac{(\bar{b} . \bar{c}) . \bar{a}}{{|\bar{a}|}^{2}}$

$\bar{a} \times (\bar{b} \times \bar{a})$

answer is B.

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Detailed Solution

$\begin{array}{l} \bar{a} - \frac{(\bar{a} \cdot \bar{b})}{| \bar{b} |^{2}} \cdot \bar{b} \\ = \frac{(\bar{b} \cdot \bar{b}) \bar{a} - (\bar{a} \cdot \bar{b}) \bar{b}}{| \bar{b} |^{2}} \\ = \frac{\bar{b} \times (\bar{a} \times \bar{b})}{| \bar{b} |^{2}} \end{array}$