1. Vectors (10)
Modelling

2. Displacement B r r r O r r r A r = 6i - 2j
An object with the position vector (6i - 2j)m is displaced by (-4i + 4j)m, what is it’s final position vector? B
s = -4i + 4j r B r A
+ s = B
= s r A B O A
= 6i - 2j r
= (6i - 2j) + (-4i + 4j) r B
= 6i - 2j - 4i + 4j r B
= (2i + 2j) m r B

3. Velocity Vector Calculation (1)s
= 5i + 5j
If a ship has a position vector 5i + 5j,
3 seconds later it has the position vector 2i - 4j
What is the average velocity in the time ? A
Firstly, what is the displacement s ? O r A
+ s = B r
s = - r A B B
= 2i - 4j
s = (2i - 4j) - (5i + 5j) B
s = 2i - 4j - 5i - 5j
s = -3i - 9j

4. Velocity Vector Calculation (2)s
= 5i + 5j
If a ship has a position vector 5i + 5j,
3 seconds later it has the position vector 2i - 4j
What is the average velocity in the time ? A
s = -3i - 9j O
average velocity
= change in position vector
time taken r B
= 2i - 4j
= (-3i - 9j) m
3 sec.
average velocity B
average velocity
= (-i - 3j) ms-1
Meaning 1 ms-1 West and 3 ms-1 South

5. Magnitude of the Velocity
average velocity
= (-i - 3j) ms-1
Magnitude of velocity
= ( )
= 10
= 3.2 ms-1 s 3 1

6. Resultant Velocity (1) A River 5ms-1 Something like this 2ms-1
If the boat sails straight across, which direction will it actually go?
A River
5ms-1
Something like this
2ms-1
Direction of flow

7. Resultant Velocity (2) 2ms-1 5ms-1 5ms-1 2ms-1
The RESULTANT VELOCITY can be found by a triangle of velocities
Resultant = 5i + 2j
2ms-1
5ms-1
The resultant
or a = = 68.2o measured from
the river bank a
2ms-1
5ms-1 
tan  = opp/adj = 2/5
 = tan-1(0.4) = 21.8o
Direction
Magnitude = ( ) = 29 = 5.4 ms-1

8. Resultant Velocity (3) Do a sketch NE Wind = 5i - 2j
A plane flys with velocity 5i - 2j
The wind blows with velocity 3i + 3j
What is the resultant velocity and direction?
= 5i - 2j
Do a sketch
= 3i + 3j
NE Wind

9. Resultant Velocity (3.5) - sketch
A plane flys with velocity 5i - 2j
The wind blows with velocity 3i + 3j
What is the resultant velocity
and direction?
resultant
velocity v
3i + 3j
Wind
5i - 2j
5i - 2j
3i + 3j
Wind
You can sketch it either
way round.
The resultant is identical
in both cases.
resultant
velocity v

10. Resultant Velocity (4) v = 3i + 3j + 5i - 2j
A plane flys with velocity 5i - 2j
The wind blows with velocity 3i + 3j
What is the resultant velocity and direction?
Wind
5i - 2j
3i + 3j
v = 3i + 3j + 5i - 2j
= 8i + j
resultant
velocity v

11. Resultant Velocity (5) N v = 8i + j
A plane flys with velocity 5i - 2j
The wind blows with velocity 3i + 3j
What is the resultant velocity and direction? N b
Bearing b =
= 82.9o
resultant
velocity v
v = 8i + j 8 1 
tan  = opp/adj = 1/8
 = tan-1(1/8) = 7.1o
Direction
Magnitude =
( ) = 65 = 8.1 ms-1

12. Modeling Techniques
A plane is 700km East and 500km North of the origin (airport).
It flies with velocity 300i - 220j km/h.
What is it’s Position Vector at time t hours.
1st: Find in terms of i and j the original position vector
of the plane. O P
50km
OP = 700i + 500j
70km

13. Modeling Techniques
A plane is 700km East and 500km North of the origin (airport).
It flies with velocity 300i - 220j km/h.
What is it’s Position Vector at time t hours. P
motion
OP = 700i + 500j O

14. 1 hour later Modeling Techniques
A plane is 700km East and 500km North of the origin (airport).
It flies with velocity 300i - 220j km/h.
What is it’s Position Vector at time t hours. P
PC = 300i - 220j
OP = 700i + 500j C O
OC = 700i + 500j + 300i - 220j
OC = 1000i + 280j
1 hour later

15. 2 hours later Modeling Techniques
A plane is 700km East and 500km North of the origin (airport).
It flies with velocity 300i - 220j km/h.
What is it’s Position Vector at time t hours. P
PC = 600i - 440j
OP = 700i + 500j C O
OC = 700i + 500j + 600i - 440j
OC = 1300i + 60j
2 hours later

16. 5 hours later Modeling Techniques
A plane is 700km East and 500km North of the origin (airport).
It flies with velocity 300i - 220j km/h.
What is it’s Position Vector at time t hours. P
OP = 700i + 500j
PC = 5(300i - 440j) O
OC = 700i + 500j i j
OC = 2200i - 700j C
5 hours later

17. t hours later Modeling Techniques
A plane is 700km East and 500km North of the origin (airport).
It flies with velocity 300i - 220j km/h.
What is it’s Position Vector at time t hours. P
PC = t(300i - 220)j
OP = 700i + 500j
Distance = speed x time C O
OC = 700i + 500j + t(300i - 220j)
Original
Point
Parallel
Vector
t hours later

18. [ ] Modeling A Problem: example - 1 Exercise 17D: ‘Page 407’ Q 3
An Ocean Liner is at (6, -6) is cruising at 10 km/h in
the direction A fishing boat is anchored at (0,0). -3 4
[ ]
L (6,-6)
F (0,0) O
A) Find in terms of i and j the original position vector
of the liner from the fishing boat
OL = 6i - 6j
Exercise 17D: ‘Page 407’ Q 3

19. [ ] [ ] [ ] [ ] Modeling A Problem - 2
An Ocean Liner is at (6, -6) is cruising at 10 km/h in
the direction A fishing boat is anchored at (0,0). -3 4
[ ] L
F (0,0) O
B) Find the position vector of the liner at time t -3 4
[ ]
is the direction of the velocity vector
Magnitude =
((-3)2 + 42) = 25 = 5
Velocity is 10 km/h : v = = -3 4
[ ]
(twice as long) -6 8
In t hours; moved by t x -6 8
[ ]
OL = 6i - 6j + t(-6i + 8j)

20. Modeling A Problem - 3
An Ocean Liner is at (6, -6) is cruising at 10 km/h in
the direction A fishing boat is anchored at (0,0). -3 4
[ ] L
F (0,0) O
OL = 6i - 6j + t(-6i + 8j)
C) Find the time t when the liner is due East of the fishing boat
Due East when j component is zero
OL = 6i - 6j + t(-6i + 8j)
(-6 + 8t)j = 0
-6 + 8t = 0
8t = 6
t = 6/8
At time 3/4 hour