1. Vectors a a ~ ~ Some terms to know: Vectors:
Lines with length (magnitude) and direction O A
Vector a ~ a ~

2. Vectors a ~ | | Zero vector or Null vector No length No direction
Magnitude of is represented by a ~
| |
Or magnitude of is represented by
Direction is represented by column vector
Zero vector
or Null vector
No length
No direction

3. Vectors Q y = +4 P x = +2 Movement parallel to the x-axis
to the y-axis P Q
y = +4
x = +2

4. Vectors P Q
y = - 4
x = - 2

5. Equal vectors or equivalent vectorsP Q
Same length R S
Same direction

6. reverses the direction
Vectors
Negative vectors P Q
Same length R S
opposite direction
NOTE: Negative sign
reverses the direction

7. Let’s do up Exercise 5A in your foolscap paper
Q. 1, 3, 5

8. Vectors
Vector addition
Vector subtraction

9. Triangle law of vector addition Ending point
Vectors
Triangle law of vector addition R
Ending point P Q
Starting point
Parallelogram law of vector addition P Q R
Ending point
Starting point S

10. Let’s do up Exercise 5B in your foolscap paper
Q. 3, 4, 6, 7

11. Vectors
Scalar multiplication

12. Let’s do up Exercise 5C in your foolscap paper
Q. 1, 5, 10

13. Vectors P Q
y = - 5
x = 2

14. Vectors
Parallel vectors P Q
Same gradient R S

15. Vectors
Gradient of vector P Q
y = +4
x = +2

16. Vectors Collinear vectors Lie on the same straight line
P, Q and R are collinear
=> P, Q and R lie on the same straight line R P P Q

17. Vectors
Eg. Given c = , find
(a) |2c| (b) |-2c| (c) 2|c| ~
(a) 2c = 2
|2c| =

18. Vectors
Eg. Given c = , find
(a) |2c| (b) |-2c| (c) 2|c| ~
(b) -2c = -2
|-2c| =

19. Vectors
Eg. Given c = , find
(a) |2c| (b) |-2c| (c) 2|c| ~
(c) 2|c| =

20. Therefore, |2c| = |-2c| = 2|c|
Vectors
Notice:
|2c| = 10 units
|-2c| = 10 units
2|c| = 10 units
Therefore, |2c| = |-2c| = 2|c|
In conclusion:
|kc| = |-kc| = k|c|, where k is any positive number

21. Vectors
Eg. Given a = and b = , find
(a) |a| + |b| (b) |a+b| ~ ~
|a| + |b|
a+b
|a+b| =
Note: |a| + |b| ≠ |a + b|

22. Vectors
Eg. Given a = and b = , find
(c) |2b-a| (d) 2|b| + |a| ~ ~
2b-a
2|b| + |a|
|2b-a| =
Note: |2b -a| ≠ 2|b| - |a|

23. Let’s do up Exercise 5D in your foolscap paper
Q. 1, 6, 8

24. Vectors Position vectors: always with respect to origin
Can you tell me what is the position vector of P? x y O P

25. Vectors In general, position vector of any point, say P, is given by yx y O P

26. Vectors
Position vectors y P Q O x

27. Let’s do up Exercise 5E in your foolscap paper
Q. 3, 6, 12