OABC is a trapezium. D is the point on OB such that OD : DB = 2:3

E is the point on BC such that BE : EC = 1:4

Work out the vector of DE in terms of a & b. Give your answer in its simplest form.

DE = 2/5a + b

Step-by-step explanation:

Find OB:

OB = a + b

Find BC:

OB + BC = OCa + b + BC = 3bBC = 2b - a

Find DB:

OD/DB = 2/3 ⇒ OD + DB = OB ⇒ 2x + 3x = OB ⇒ 5x = OB ⇒ x = OB/5DB = 3x = 3/5 OB = 3/5(a + b)

Find BE:

BE/EC = 1/4 ⇒ BE + EC = BC ⇒ x + 4x = BC ⇒ 5x = BC ⇒ x = 1/5 BCBE = x ⇒ BE = 1/5 BC ⇒ BE = 1/5(2b - a)

Find DE:

DE = DB + BEDE = 3/5(a + b) +  1/5(2b - a) DE = 1/5(3a + 3b + 2b - a)DE = 1/5(2a + 5b)DE = 2/5a + b