a→ and 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 a→ is given by:
a→ ⋅ b→|a→|2 a→ − b→
a→ ⋅ b→|b→|2 b→ − a→
b→ × (a→ × b→)|b→|2
a→ ⋅ b→|a→|2 a→ + b→
The altitude required is BN→
BN→=ON→−OB→
= projection of OB→ on ON→−b→
= (b→ ⋅ a∧) a∧ − b→
= b→ ⋅ a→|a→| a→|a→| − b→ = (b→ ⋅a→) a→|a→|2 − b→