Pentagon problem involving vectors

[Guest]

Let ABCDE (counterclockwise) be a regular pentagon. Express AD, AE and BE in base (AB, AC).

[Guest]

[tex]AD = -\phi AB+\phi AC[/tex]
[tex]AE = -\phi AB+AC[/tex]
[tex]BE = -(1+\phi) AB+AC[/tex]
where [tex]\phi = \frac{1+\sqrt{5}}{2}[/tex].

(The trick is to notice various vectors are parallel like AC and ED and the ratio of their lengths is [tex]\phi[/tex]).

Hope this helps,

R. Baber.