Question 1:
Given that O (0, 0), A (–3, 4) and B (–9, 12), find in terms of the unit vectors, i˜ and j˜.
Solution:
(a)
A=(−3,4), thus →OA=−3i˜+4j˜B=(−9,12), thus →OB=−9i˜+12j˜→AB=→AO+→OB→AB=−(−3i˜+4j˜)+(−9i˜+12j˜)→AB=3i˜−4j˜−9i˜+12j˜→AB=−6i˜+8j˜
(b)
The magnitude of |→AB|,|→AB|=√(−6)2+(8)2=10∴The unit vector in the direction of →AB,→AB|→AB|=110(−6i˜+8j˜)=−35i˜+45j˜
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+→OB→AB=−(−3i˜+4j˜)+(−9i˜+12j˜)→AB=3i˜−4j˜−9i˜+12j˜→AB=−6i˜+8j˜
(b)
The magnitude of |→AB|,|→AB|=√(−6)2+(8)2=10∴The 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 2→AB+→BC=(12−3)
Solution:
A=(−32), B=(46) and C=(mn)→AB=→AO+→OB→AB=−(−32)+(46)=(74)→BC=→BO+→OC→BC=−(46)+(mn)=(−4+m−6+n)Given 2→AB+→BC=(12−3)2(74)+(−4+m−6+n)=(12−3)(14−4+m8−6+n)=(12−3)10+m=12m=22+n=−3n=−5
Given that A (–3, 2), B (4, 6) and C (m, n), find the value of m and of n such that 2→AB+→BC=(12−3)
Solution:
A=(−32), B=(46) and C=(mn)→AB=→AO+→OB→AB=−(−32)+(46)=(74)→BC=→BO+→OC→BC=−(46)+(mn)=(−4+m−6+n)Given 2→AB+→BC=(12−3)2(74)+(−4+m−6+n)=(12−3)(14−4+m8−6+n)=(12−3)10+m=12m=22+n=−3n=−5
Question 3:
Solution:
→OB=→OA+→AB=3x˜+12y˜→OD=3→DB→OD→DB=31→OD:→DB=3:1∴→OD=34→OB=34(3x˜+12y˜)=94x˜+9y˜
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=3→DB→OD→DB=31→OD:→DB=3:1∴→OD=34→OB=34(3x˜+12y˜)=94x˜+9y˜