4.7.1 Vectors, Long Questions (Question 1 & 2)


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 OB, AQ= 1 4 AB,  OP =4 b ˜  and  OA =8 a ˜ .  

(a) Express in terms of   a ˜  and/ or  b ˜ :
( i ) A P (ii) O Q

(b)(i) Given that A R = h A P , state   A R  in terms of h   a ˜  and  b ˜ .
 (ii) Given that   R Q = k O Q , state  in terms of k,   a ˜  and  b ˜ .

(c) Using   A Q = A R + R Q ,   find the value of h and of k.


Solution
:

(a)(i)
A P = A O + O P A P = O A + O P A P = 8 a ˜ + 4 b ˜


(a)(ii)
O Q = O A + A Q O Q = 8 a ˜ + 1 4 A B O Q = 8 a ˜ + 1 4 ( A O + O B ) O Q = 8 a ˜ + 1 4 ( 8 a ˜ + 4 O P ) O Q = 8 a ˜ + 1 4 ( 8 a ˜ + 4 ( 4 b ˜ ) ) O Q = 8 a ˜ 2 a ˜ + 4 b ˜ O Q = 6 a ˜ + 4 b ˜


(b)(i)
A R = h A P A R = h ( 8 a ˜ + 4 b ˜ ) A R = 8 h a ˜ + 4 h b ˜



(b)(ii)
R Q = k O Q R Q = k ( 6 a ˜ + 4 b ˜ ) R Q = 6 k a ˜ + 4 k b ˜


(c)
A Q = A R + R Q A Q = 8 h a ˜ + 4 h b ˜ + ( 6 k a ˜ + 4 k b ˜ ) A O + O Q = 8 h a ˜ + 4 h b ˜ + 6 k a ˜ + 4 k b ˜ 8 a ˜ + 6 a ˜ + 4 b ˜ = 8 h a ˜ + 6 k a ˜ + 4 h b ˜ + 4 k b ˜ 2 a ˜ + 4 b ˜ = 8 h a ˜ + 6 k a ˜ + 4 h b ˜ + 4 k b ˜ 2 = 8 h + 6 k 1 = 4 h + 3 k ( 1 ) 4 = 4 h + 4 k 1 = h + k k = 1 h ( 2 ) Substitute (2) into (1), 1 = 4 h + 3 ( 1 h ) 1 = 4 h + 3 3 h 4 = 7 h h = 4 7 From (2), k = 1 4 7 = 3 7


Question 2:
Given that   A B = ( 10 14 ) , O B = ( 4 6 ) and C D = ( m 7 ) , find
(a) the coordinates of A,
(b) the unit vector in the direction of O A .
(c) the value of m if CD is parallel to AB .

Solution:

(a)
A B = A O + O B ( 10 14 ) = ( x y ) + ( 4 6 ) ( x y ) = ( 10 14 ) ( 4 6 ) A O = ( 6 8 ) O A = ( 6 8 ) A = ( 6 , 8 )



(b)
| OA |= ( 6 ) 2 + ( 8 ) 2 | OA |= 100 =10 the unit vector in the direction of  OA = OA | OA | = ( 6 8 ) 10 = 1 10 ( 6 8 ) =( 3 5 4 5 )


(c)
Given  CD  parallel  AB   CD =k AB ( m 7 )=k( 10 14 ) ( m 7 )=( 10k 14k ) 7=14k k= 1 2 m=10k=10( 1 2 )=5

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