$\vec{a} and \vec{b} are two given vectors. On these vectors as adjacent sides, a parallelogram$ $is constructed. The vector which is the altitude of the parallelogram and perpendicular$ $to \vec{a} is given by:$

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$(\frac{\vec{a} \cdot \vec{b}}{| \vec{a} |^{2}}) \vec{a} + \vec{b}$

$(\frac{\vec{a} \cdot \vec{b}}{| \vec{b} |^{2}}) \vec{b} - \vec{a}$

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

$(\frac{\vec{a} \cdot \vec{b}}{| \vec{a} |^{2}}) \vec{a} - \vec{b}$

answer is A.

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Detailed Solution

$The altitude required is \vec{B N}$

$\vec{B N} = \vec{O N} - \vec{O B}$

$= projection of \vec{O B} on \vec{O N} - \vec{b}$

$= (\vec{b} \cdot \overset{\land}{a}) \overset{\land}{a} - \vec{b}$

$= (\vec{b} \cdot \frac{\vec{a}}{| \vec{a} |}) \frac{\vec{a}}{| \vec{a} |} - \vec{b} = \frac{(\vec{b} \cdot \vec{a}) \vec{a}}{| \vec{a} |^{2}} - \vec{b}$

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