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

OP= 1 4 OBAQ= 1 4 AB OP =4 b ˜  and  OA =8 a ˜ . 

(a) Express in terms of   a ˜  and/ or  b ˜ :
(i)  AP (ii)  OQ

(b)(i) Given that AR =h AP , state  AR    in terms of h  a ˜  and  b ˜ .
     (ii) Given that   RQ =k OQ , state  in terms of k,   a ˜  and  b ˜ .

(c) Using   AQ = AR + RQ , find the value of h and of k.

Solution:
(a)(i)
AP = AO + OP AP = OA + OP AP =8 a ˜ +4 b ˜
(a)(ii)
OQ = OA + AQ OQ =8 a ˜ + 1 4 AB OQ =8 a ˜ + 1 4 ( AO + OB ) OQ =8 a ˜ + 1 4 ( 8 a ˜ +4 OP ) OQ =8 a ˜ + 1 4 ( 8 a ˜ +4( 4 b ˜ ) ) OQ =8 a ˜ 2 a ˜ +4 b ˜ OQ =6 a ˜ +4 b ˜
(b)(i)
AR =h AP AR =h( 8 a ˜ +4 b ˜ ) AR =8h a ˜ +4h b ˜
(b)(ii)
RQ =k OQ RQ =k( 6 a ˜ +4 b ˜ ) RQ =6k a ˜ +4k b ˜
(c)
AQ = AR + RQ AQ =8h a ˜ +4h b ˜ +( 6k a ˜ +4k b ˜ ) AO + OQ =8h a ˜ +4h b ˜ +6k a ˜ +4k b ˜ 8 a ˜ +6 a ˜ +4 b ˜ =8h a ˜ +6k a ˜ +4h b ˜ +4k b ˜ 2 a ˜ +4 b ˜ =8h a ˜ +6k a ˜ +4h b ˜ +4k b ˜ 2=8h+6k 1=4h+3k(1) 4=4h+4k 1=h+k k=1h(2) Substitute (2) into (1), 1=4h+3( 1h ) 1=4h+33h 4=7h h= 4 7 From (2), k=1 4 7 = 3 7