# Vector Long Questions (Question 3)

Question 3:
In diagram below, PQRS is a quadrilateral. PTS and TUR are straight lines.

It is given that
(a) Express in terms of  $\underset{˜}{x}$ and/or  $\underset{˜}{y}$ :
(i)  $\stackrel{\to }{QS}$
(ii)  $\stackrel{\to }{TR}$
(b) Show that  the points  Q, U and S  are collinear.
(c) If   = 2  and = 3, find  $|\stackrel{\to }{QS}|$

Solution:
(a)(i)

(a)(ii)
$\begin{array}{l}\stackrel{\to }{TR}=\stackrel{\to }{TS}+\stackrel{\to }{SR}\\ \stackrel{\to }{TR}=\frac{3}{4}\stackrel{\to }{PS}+25\underset{˜}{x}-24\underset{˜}{y}\\ \stackrel{\to }{TR}=\frac{3}{4}\left(32\underset{˜}{y}\right)+25\underset{˜}{x}-24\underset{˜}{y}\\ \stackrel{\to }{TR}=24\underset{˜}{y}+25\underset{˜}{x}-24\underset{˜}{y}\\ \stackrel{\to }{TR}=25\underset{˜}{x}\end{array}$
(b)

(c)
$\begin{array}{l}\stackrel{\to }{PS}=32\underset{˜}{y}\\ |\stackrel{\to }{PS}|=32|\underset{˜}{y}|\\ |\stackrel{\to }{PS}|=32×3=96\\ \\ \stackrel{\to }{PQ}=20\underset{˜}{x}\\ |\stackrel{\to }{PQ}|=20|\underset{˜}{x}|\\ |\stackrel{\to }{PQ}|=20×2=40\\ \\ \therefore |\stackrel{\to }{QS}|=\sqrt{{96}^{2}+{40}^{2}}\\ |\stackrel{\to }{QS}|=104\end{array}$