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CH-24-001 · Vector division and a parallel-line proof

OAB is a triangle. OA→=a\overrightarrow{OA}=\mathbf a and OB→=b\overrightarrow{OB}=\mathbf b. D lies on AB with AD:DB = 10:13. E lies on OA with OE:EA = 13:10.
OABDEba
Diagram not accurately drawn

(a) Express OD and ED as vectors in terms of a and b. Give your answers in their simplest form.

(b) Hence prove that ED is parallel to OB.

Part (a) Vectors: Ratio Division & Parallel Line Proof Booklet p. 90
Method
1) Write routes AB = b − a, OD = OA + frac × AB, ED = EO + OD
2) Simplify to show ED is a scalar multiple of OB ⇒ parallel
Worked Example (CH-24-001 · Part (a))
Express OD and ED as vectors in terms of a and b. Give your answers in their simplest form.
1)
AB→=b−a\overrightarrow{AB}=\mathbf b-\mathbf a
2)
Use OD→=a+1023(b−a)\overrightarrow{OD}=\mathbf a+\frac{10}{23}(\mathbf b-\mathbf a).
3)
OD→=1323a+1023b\overrightarrow{OD}=\frac{13}{23}\mathbf a+\frac{10}{23}\mathbf b
4)
Subtract OE→=1323a\overrightarrow{OE}=\frac{13}{23}\mathbf a.
5)
ED→=1023b\overrightarrow{ED}=\frac{10}{23}\mathbf b
Ans = OD→=1323a+1023b\overrightarrow{OD}=\frac{13}{23}\mathbf a+\frac{10}{23}\mathbf b; ED→=1023b\overrightarrow{ED}=\frac{10}{23}\mathbf b
Part (b) Vectors: Ratio Division & Parallel Line Proof Booklet p. 90
Method
1) Write routes AB = b − a, OD = OA + frac × AB, ED = EO + OD
2) Simplify to show ED is a scalar multiple of OB ⇒ parallel
Worked Example (CH-24-001 · Part (b))
Hence prove that ED is parallel to OB.
1)
The vector ED is a non-zero scalar multiple of OB, so ED is parallel to OB.
Ans = ED is a non-zero scalar multiple of OB, so the lines are parallel.