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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) AB
(b) the unit vector in the direction of  AB

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


(b)
The magnitude of |AB|,|AB|=(6)2+(8)2=10The unit vector in the direction of AB,AB|AB|=110(6i˜+8j˜)=35i˜+45j˜



Question 2:
Given that A (3, 2), B (4, 6) and C (m, n), find the value of m and of n such that    2AB+BC=(123)

Solution:

A=(32), B=(46) and C=(mn)AB=AO+OBAB=(32)+(46)=(74)BC=BO+OCBC=(46)+(mn)=(4+m6+n)Given 2AB+BC=(123)2(74)+(4+m6+n)=(123)(144+m86+n)=(123)10+m=12m=22+n=3n=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:

OB=OA+AB=3x˜+12y˜OD=3DBODDB=31OD:DB=3:1OD=34OB=34(3x˜+12y˜)=94x˜+9y˜

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