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8.6.1 Vectors, Short Questions (Question 1 – 3)


Question 1:
Given that O (0, 0), A (–3, 4) and B (–9, 12), find in terms of the unit vectors,   i ˜ and   j ˜.
(a) A B →
(b) the unit vector in the direction of  A B →

Solution:
(a) 
A=(−3,4), thus  OA → =−3 i ˜ +4 j ˜ B=(−9,12), thus  OB → =−9 i ˜ +12 j ˜ AB → = AO → + OB → AB → =−( −3 i ˜ +4 j ˜ )+( −9 i ˜ +12 j ˜ ) AB → =3 i ˜ −4 j ˜ −9 i ˜ +12 j ˜ AB → =−6 i ˜ +8 j ˜


(b)
The magnitude of | AB → |, | AB → |= ( −6 ) 2 + ( 8 ) 2 =10 ∴The unit vector in the direction of  AB → , AB → | AB → | = 1 10 ( −6 i ˜ +8 j ˜ )=− 3 5 i ˜ + 4 5 j ˜



Question 2:
Given that A (–3, 2), B (4, 6) and C (m, n), find the value of m and of n such that    2 A B → + B C → = ( 12 − 3 )

Solution:
A=( −3 2 ), B=( 4 6 ) and C=( m n ) AB → = AO → + OB → AB → =−( −3 2 )+( 4 6 )=( 7 4 ) BC → = BO → + OC → BC → =−( 4 6 )+( m n )=( −4+m −6+n ) Given 2 AB → + BC → =( 12 −3 ) 2( 7 4 )+( −4+m −6+n )=( 12 −3 ) ( 14−4+m 8−6+n )=( 12 −3 ) 10+m=12 m=2 2+n=−3 n=−5




Question 3:
Diagram below shows a rectangle OABC and the point D lies on the straight line OB.
 
It is given that OD = 3DB.
Express  OD →  in terms of  x ˜  and  y ˜ .

Solution:
O B → = O A → + A B → = 3 x ˜ + 12 y ˜ O D → = 3 D B → O D → D B → = 3 1 O D → : D B → = 3 : 1 ∴ O D → = 3 4 O B → = 3 4 ( 3 x ˜ + 12 y ˜ ) = 9 4 x ˜ + 9 y ˜

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