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 | A B | , | A B | = ( 6 ) 2 + ( 8 ) 2 = 10 The unit vector in the direction of A B , A B | A B | = 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 ) ( 144+m 86+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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