question_answer

If vectors \[\vec{a}\] and \[\vec{b}\] are two adjacent sides of a Parallelogram, then the vector representing the altitude of the parallelogram which is perpendicular to \[\vec{a}\] is

A) \[\vec{b}+\frac{\vec{b}\times \vec{a}}{{{\left| {\vec{a}} \right|}^{2}}}\]         

B) \[\frac{\vec{a}\cdot \vec{b}}{{{\left| {\vec{b}} \right|}^{2}}}\]

C) \[\vec{b}-\frac{\vec{b}\cdot \vec{a}}{{{\left| {\vec{a}} \right|}^{2}}}\vec{a}\]

D) \[\frac{\vec{a}\times (\vec{b}\times \vec{a})}{{{\left| {\vec{b}} \right|}^{2}}}\]

Correct Answer: C

Solution :

[c] Let \[\overrightarrow{OD}=t\vec{a}\] \[\therefore \overrightarrow{OD}=\vec{b}-t\vec{a}\] Or \[(\vec{b}-t\vec{a}).\vec{a}=0\]\[(\therefore \overrightarrow{DB}\bot \overrightarrow{OA})\] Or \[t=\frac{\vec{b}\cdot \vec{a}}{{{\left| {\vec{a}} \right|}^{2}}}\]