a→ and b→ are given given vectors. with these vectors as adjacent sides, a parallelogram is constructed. The vector which is the altitude of the parallelogram  and which is perpendicular to a→ is

We haveAM= projection of b→ on a→=a→⋅b→|a→|∴ AM→=a→⋅b→|a→|2a→ Now, in △ADMAD→=AM→+MD→ or DM→=AM→−AD→⇒DM→=(a→⋅b→)a→|a→|2−b→=1|a→|2(a→⋅b→)a→−|a→|2b→⇒MD→=1|a→|2|a→|2b→−(a→⋅b→)a→now, a→×(a→×b→)|a→|2=1|a→|2[(a→⋅b→)a→−(a→⋅a→)b→]=DM→