Vectors & + how to demonstrate that lines are parallels

[fb34090]

Hello, I need your help to finish some maths homeworks. I worked hard on it but still have problems to solve c & d questions

For the triangle ABC, points D, E, F et G are defined as follows :

http://www.cjoint.com/c/EJtoVY2uAyQ

1) 3[tex]\vec{DB}[/tex] + 7[tex]\vec{DC}[/tex] = 0
2) E is the middle of [AC]
3) 3[tex]\vec{FB}[/tex] + [tex]\vec{FA}[/tex] = 0
4)G is symmetrical to E with respect to A

Demonstrate that (ED) and (FG) are parallels.


a) Justify that [tex]\vec{CD}[/tex] = 3/10[tex]\vec{CB}[/tex] and that [tex]\vec{AF}[/tex] = 3/4[tex]\vec{AB}[/tex]
b) Using the Chasles relation, express [tex]\vec{FG }[/tex] according to [tex]\vec{AB}[/tex] and [tex]\vec{AC}[/tex]
c) Do the same for [tex]\vec{ED}[/tex] according to [tex]\vec{AB}[/tex] and [tex]\vec{AC}[/tex]
d) Conclude that (ED) and (FG) are parallels

Here's what I have done :

a)
3[tex]\vec{FB}[/tex]+[tex]\vec{FA}[/tex]=0
3[tex](\vec{FA}[/tex]+[tex]\vec{AB}[/tex])+[tex]\vec{FA}[/tex]=0
4[tex]\vec{FA}[/tex]+3[tex]\vec{AB}[/tex]=0
-4[tex]\vec{FA}[/tex]=3[tex]\vec{AB}[/tex]
[tex]\vec{AF}[/tex]=3/4[tex]\vec{AB}[/tex]

b)
[tex]\vec{FG}[/tex] = [tex]\vec{FA}[/tex] + [tex]\vec{AG}[/tex]
[tex]\vec{FG}[/tex] = -[tex]\vec{AF}[/tex] - [tex]\vec{GA}[/tex]
[tex]\vec{FG}[/tex] = -3/4[tex]\vec{AB}[/tex] - 1/2 [tex]\vec{AC}[/tex]

As E is symetrical of G with respect to A :
A is the middle [GE]
Therefore
[GA] = [AE]
E is the middle of [AC] thus [AE] = [EC]
and [AE]=[EC]=[GA]
So
[tex]\vec{AE}[/tex] + [tex]\vec{EC}[/tex] = [tex]\vec{AC}[/tex]
and [tex]\vec{GA}[/tex] = 1/2 [tex]\vec{AC}[/tex]

c) It's where i'm encountering problems :
Here's what i've done
[tex]\vec{ED}=\vec{EA}[/tex]+[tex]\vec{AB}[/tex]+ [tex]\vec{BD}[/tex]
1/2[tex]\vec{CA}[/tex] + [tex]\vec{AB}[/tex] + ([tex]\vec{BA}[/tex]+[tex]\vec{AC}[/tex]+[tex]\vec{CD})[/tex]
1/2[tex]\vec{CA}[/tex] + [tex]\vec{AB}[/tex] - [tex]\vec{AB}[/tex] +[tex]\vec{AC}[/tex]+[tex]\vec{CD}[/tex]
1/2[tex]\vec{CA}[/tex]+ [tex]\vec{AC}[/tex] + [tex]\vec{CD}[/tex]
1/2[tex]\vec{CA}[/tex]+ [tex]\vec{AC}[/tex] - [tex]\vec{DC}[/tex]
... ???

d) I have no idea how to conclude / parallels

I'd appreciate your help, not giving me the answers exactly but helping me to find them...

[Guest]

You are doing pretty good.

For part (c) you can continue from where you got to.
You need some way of converting [tex]\vec{CD}[/tex] (or [tex]\vec{BD}[/tex]) into some combination of [tex]\vec{AB}[/tex] and [tex]\vec{AC}[/tex].
The trick is to realise it is just some multiple of the vector [tex]\vec{CB}[/tex] (in particular from part (a) we know that [tex]\vec{CD}=\frac{3}{10}\vec{CB}[/tex]) and that [tex]\vec{CB}[/tex] can be easily written in terms of [tex]\vec{AB}[/tex] and [tex]\vec{AC}[/tex] (look at the triangle [tex]ABC[/tex] and it should be obvious what the relation is).

For part (d): Two vectors are parallel if they are multiples of each other. In particular, for this question if you can find some scalar [tex]\lambda[/tex] such that the relation [tex]\vec{ED}=\lambda\times\vec{FG}[/tex] holds, then the two vectors must be parallel.

Hope this helped,

R. Baber