I am trying to solve the following problem, and I have a solution which I don't understand, can you help me?
"Show that in a triangle the perpendiculars drawn from the vertices are concurrent. "

Question

I am trying to solve the following problem, and I have a solution which I don't understand, can you help me?
"Show that in a triangle the perpendiculars drawn from the vertices are concurrent. "

2bornot2b · Answer

Here is the solution that I don't understand

mani_jha · Answer

Exactly which part u dont understand? It is all about triangle law of vector addition and dot products

2bornot2b · Answer

What has the solution shown so that its clear that the perpendiculars are concurrent.

mani_jha · Answer

Any two lines have to intersect(unless they are parallel). Now, if a third line passes through the intersection of these two lines, then the three are concurrent. In this solution, they have taken two perpendiculars. Through their intersection point a line has been drawn, and if that line happens to be the altitude through C, then all perpendiculars have to be concurrent.
That's what we have to prove here. The dot product of two perpendicular lines is 0. So, we must prove that the dot product of AB and CF is zero.
I hope it helped, if not please say

2bornot2b · Answer

Just a sec, let me read it and understand

2bornot2b · Answer

Have you seen they have used a constant like $l$ while writing
$$la(c-b)=0$$
Whats the need of that $l$

mani_jha · Answer

Well, let me guess. Here OA=a. But actually we see that AD is perpendicular to BC and also intersects it. So, they took the length of AD instead of AO. So, AD/AO=l is assumed. so that al is the length of AD. 
But I dont think there is any compulsion of taking l and m here.

2bornot2b · Answer

Do you have any better solution for it?

mani_jha · Answer

Of course there is a geometrical solution to this. Well, let me work on it. I will post it as soon as i complete it

2bornot2b · Answer

No thank you, I need to make it through vector

2bornot2b · Answer

Thats mandatory for me

2bornot2b · Answer

OK, thank you!

2bornot2b · Answer

OK so mani jha provided with the explanation that " The dot product of two perpendicular lines is 0. So, we must prove that the dot product of AB and CF is zero.
I hope it helped, if not please say", but it didn't quite satisfy me. Can you think of any better explanation. 
Does this solution have anything to do with vector equation?