1.5: Intersecting lines using vectors

4PM1/1/June/2021 — Question 10 · 14 marks

1.5 diagram 1
Figure 1 shows triangle OAB\displaystyle OAB and triangle OCD\displaystyle OCD.
OA→=5p,AB→=3q,OC→=32OB→,OD→=35OA→.\overrightarrow{OA}=5\mathbf{p}, \qquad \overrightarrow{AB}=3\mathbf{q}, \qquad \overrightarrow{OC}=\frac32\overrightarrow{OB}, \qquad \overrightarrow{OD}=\frac35\overrightarrow{OA}.
(a) Find DC→\displaystyle \overrightarrow{DC} as a simplified expression in terms of p\displaystyle \mathbf{p} and q\displaystyle \mathbf{q}.
(3)
The line DC\displaystyle DC meets the line AB\displaystyle AB at F\displaystyle F.
(b) Using a vector method, find OF→\displaystyle \overrightarrow{OF} as a simplified expression in terms of p\displaystyle \mathbf{p} and q\displaystyle \mathbf{q}.
(7)
The point G\displaystyle G lies on OB\displaystyle OB such that FG\displaystyle FG is parallel to AO\displaystyle AO.
(c) Using a vector method, find OG→\displaystyle \overrightarrow{OG} as a simplified expression in terms of p\displaystyle \mathbf{p} and q\displaystyle \mathbf{q}.
(4)