## 2bornot2b 3 years ago 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. "

1. 2bornot2b

Here is the solution that I don't understand

2. Mani_Jha

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

3. 2bornot2b

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

4. Mani_Jha

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

5. 2bornot2b

Just a sec, let me read it and understand

6. 2bornot2b

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

7. Mani_Jha

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.

8. 2bornot2b

Do you have any better solution for it?

9. Mani_Jha

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

10. 2bornot2b

No thank you, I need to make it through vector

11. 2bornot2b

Thats mandatory for me

12. 2bornot2b

OK, thank you!

13. 2bornot2b

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?