# Vectors Long Questions (Question 1)

Question 1:

The above diagram shows triangle OAB. The straight line AP intersects the straight line OQ at R. It is given that

(a) Express in terms of

(b)(i) Given that$\stackrel{\to }{AR}=h\stackrel{\to }{AP}$ , state in terms of h
(ii) Given that  $\stackrel{\to }{RQ}=k\stackrel{\to }{OQ},$ state  in terms of k,

(c) Using  $\stackrel{\to }{AQ}=\stackrel{\to }{AR}+\stackrel{\to }{RQ},$ find the value of h and of k.

Solution:
(a)(i)
$\begin{array}{l}\stackrel{\to }{AP}=\stackrel{\to }{AO}+\stackrel{\to }{OP}\\ \stackrel{\to }{AP}=-\stackrel{\to }{OA}+\stackrel{\to }{OP}\\ \stackrel{\to }{AP}=-8\underset{˜}{a}+4\underset{˜}{b}\end{array}$
(a)(ii)
$\begin{array}{l}\stackrel{\to }{OQ}=\stackrel{\to }{OA}+\stackrel{\to }{AQ}\\ \stackrel{\to }{OQ}=8\underset{˜}{a}+\frac{1}{4}\stackrel{\to }{AB}\\ \stackrel{\to }{OQ}=8\underset{˜}{a}+\frac{1}{4}\left(\stackrel{\to }{AO}+\stackrel{\to }{OB}\right)\\ \stackrel{\to }{OQ}=8\underset{˜}{a}+\frac{1}{4}\left(-8\underset{˜}{a}+4\stackrel{\to }{OP}\right)\\ \stackrel{\to }{OQ}=8\underset{˜}{a}+\frac{1}{4}\left(-8\underset{˜}{a}+4\left(4\underset{˜}{b}\right)\right)\\ \stackrel{\to }{OQ}=8\underset{˜}{a}-2\underset{˜}{a}+4\underset{˜}{b}\\ \stackrel{\to }{OQ}=6\underset{˜}{a}+4\underset{˜}{b}\end{array}$
(b)(i)
$\begin{array}{l}\stackrel{\to }{AR}=h\stackrel{\to }{AP}\\ \stackrel{\to }{AR}=h\left(-8\underset{˜}{a}+4\underset{˜}{b}\right)\\ \stackrel{\to }{AR}=-8h\underset{˜}{a}+4h\underset{˜}{b}\end{array}$
(b)(ii)
$\begin{array}{l}\stackrel{\to }{RQ}=k\stackrel{\to }{OQ}\\ \stackrel{\to }{RQ}=k\left(6\underset{˜}{a}+4\underset{˜}{b}\right)\\ \stackrel{\to }{RQ}=6k\underset{˜}{a}+4k\underset{˜}{b}\end{array}$
(c)