Vector Long Questions (Question 3)


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


It is given that   PQ =20 x ˜ ,   PT =8 y ˜ ,   SR =25 x ˜ 24 y ˜ ,   PT = 1 4 PS   and   TU = 3 5 TR
(a) Express in terms of   x ˜ and/or   y ˜ :
     (i)   QS
     (ii)   TR
(b) Show that  the points  Q, U and S  are collinear.
(c) If   | x ˜ | = 2  and  | y ˜ | = 3, find   | QS |

Solution:
(a)(i)
QS = QP + PS QS =20 x ˜ +32 y ˜ Given  PT = 1 4 PS PS =4 PT =4( 8 y ˜ )=32 y ˜
(a)(ii)
TR = TS + SR TR = 3 4 PS +25 x ˜ 24 y ˜ TR = 3 4 ( 32 y ˜ )+25 x ˜ 24 y ˜ TR =24 y ˜ +25 x ˜ 24 y ˜ TR =25 x ˜
(b)
QU = QP + PT + TU QU =20 x ˜ +8 y ˜ + 3 5 ( 25 x ˜ ) Given  TU = 3 5 TR QU =20 x ˜ +8 y ˜ +15 x ˜ QU =5 x ˜ +8 y ˜ From (a)(i)  QS =20 x ˜ +32 y ˜ QS QU = 20 x ˜ +32 y ˜ 5 x ˜ +8 y ˜ QS QU = 4( 5 x ˜ +8 y ˜ ) ( 5 x ˜ +8 y ˜ ) QS QU =4 QS =4 QU   Q, U and S are collinear.
(c)
PS =32 y ˜ | PS |=32| y ˜ | | PS |=32×3=96 PQ =20 x ˜ | PQ |=20| x ˜ | | PQ |=20×2=40 | QS |= 96 2 + 40 2 | QS |=104