In ∆ABC,D and E divides AB and AC in the ratio of 1:2 respectively. Prove that Vector DE=1/3BC.

Answer 1

Make a sketch

If D divides AB in the ratio of 1 : 2,
then DA = 1/3 BA
similarly EA = 1/3 CA
notice I was careful in the order I wrote EA, that way writing them as vectors would be the same.
I will use the notation v(AB) for vector AB

v(DE) = v(DA) + v(AE)
= (1/3)v(BA) + (1/3)v(AC)
= (1/3)( v(BA) + v(AC) )
= 1/3 v(BC)
as required