4.7.2 Vector, Long Questions (Question 3 & 4)


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)   Q S
 (ii) T R
(b) Show that  the points  Q, U and S  are collinear.
(c) If   | x ˜ | = 2  and | y ˜ | = 3, find   | Q S |



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)
T R = T S + S R T R = 3 4 P S + 25 x ˜ 24 y ˜ T R = 3 4 ( 32 y ˜ ) + 25 x ˜ 24 y ˜ T R = 24 y ˜ + 25 x ˜ 24 y ˜ T R = 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)









P S = 32 y ˜ | P S | = 32 | y ˜ | | P S | = 32 × 3 = 96 P Q = 20 x ˜ | P Q | = 20 | x ˜ | | P Q | = 20 × 2 = 40 | Q S | = 96 2 + 40 2 | Q S | = 104


Question 4:
Diagram below shows quadrilateral ABCD. The straight line AC intersects the straight line BD at point E.

It is given that BE:ED=2:3,  AB =10 x ˜ ,  AD =25 y ˜  and  BC = x ˜ +15 y ˜ . (a) Express in terms of  x ˜  and  y ˜ , (i)  BD (ii)  AE (b) Find the ratio AE:EC.


Solution:
(a)(i)
BD = BA + AD   = AD AB   =25 y ˜ 10 x ˜

(a)(ii)
AE = AB + BE   = AB + 2 5 BD   =10 x ˜ + 2 5 ( 25 y ˜ 10 x ˜ )   =10 x ˜ + 2 5 ( 25 y ˜ 10 x ˜ )   =10 x ˜ +10 y ˜ 4 x ˜   =6 x ˜ +10 y ˜   =2( 3 x ˜ +5 y ˜ )

(b)
EC = EB + BC   = BC BE   = BC 2 3 ED   = BC 2 3 ( EA + AD )   = x ˜ +15 y ˜ 2 3 ( 6 x ˜ 10 y ˜ +25 y ˜ )   = x ˜ +15 y ˜ 2 3 ( 6 x ˜ +15 y ˜ )   = x ˜ +15 y ˜ +4 x ˜ 10 y ˜   =3 x ˜ +5 y ˜ AE EC = 2( 3 x ˜ +5 y ˜ ) 1( 3 x ˜ +5 y ˜ ) AE:EC=2:1

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