Vectors Intro

1. Block 3
Vectors Introduction

2. What is to be learned?
• What a vector is
• Some other bits and pieces about vectors

3. A vector is
Like a journey
Need
Distance
Direction
- magnitude

4. A
B
A
4
2
4
2( )B BA

4–
2–
-4
-2( ) components

5. A
B
– 3
AB

4
-3
4
( )
u
˜
u
˜
= =

6. C
D
-2
DC

4
4
-2
( )
t
t
˜
= =

7. 3
4( )u =
˜
U = (3 , 4)
x
y U
u
˜
u
˜
u
˜
components and coordinates

8. A
B
– 4
AB

2
-4
2
( )
u
˜
u = =

9. Magnitude of a Vector

10. 5
-4
5
-4( )
v
˜
v =
˜
magnitude of v =
˜
√(52
+(-4)2)
)

11. 5
-4
5
-4( )
v
˜
v =
˜
v =
˜
√(52
+(-4)2)
)| |

12. 3
-6
-6
3
( )
t
t
˜
=
=
t ?| |
t
˜
| | √((-6)2
+ 32
)
= √45

13. Vectors
• Describe a “journey”
– Distance
– Direction
• Numbers work like coordinates
– Called components
• The magnitude of vector is calculated using
pythagoras
 magnitude
a
b
( )u =
˜
| u2
| =
˜
a2
+ b2

14. A
B
4
-3
4
-3( )AB =

u
˜
= u
˜
| u2
| =
˜
42
+ (-3)2
| u |
˜
= 5

15. C
D
2
DC

-5
-5
2
( )
t
t
˜
= = =t
˜
| | √((-5)2
+ 22
)
= √29
Name the vector in two ways, write its
components and find its magnitude.
Key Question.

16. What is to be learned?
• How to add and “subtract” vectors.
• How to multiply a vector by a scalar.

17. Vector Addition

18. a + b
4
2( )a = b =
a+b
nose to tail
1
3
( )
a
b
5
5
= ( )

19. a + b ?( )a = b =( )
= ( )
4
5
2
3
( )4
5
( )2
3+
6
8
4 + 2
5 + 3
using components
laughably easy

20. Vector Subtraction

21. vector subtraction does not exist!
(or does it?)
9 – 7
= 9 + (-7)
we add the negative of the vector

22. 5
–
5
-4( )
v
˜
v =
˜
- v =
˜
–
–
4
-5
4
( )

23. a – b ?( )a = b =( )
= ( )
4
6
2
-3
-b = ( )-2
3
a + -b
( )4
6
( )-2
3
+
2
9
4 – 2
6 – (-3)

24. Multiplying by a scalar

25. 2
-1( )a = 3a
a
= a + a + a
a
a
3a =
6
-3( )
3a

26. Addition, Subtraction and
Multiplication (by a Scalar)
Normal Number

27. a + b
5
2( )a = b =
a+b
nose to tail
1
3
( )
a
b
6
5
= ( )
adding

28. a – b ?( )a = b =( )
= ( )
7
6
4
1
3
5
subtracting Theory – Add the negative
Practice -

29. Scalar Multiplication
( )a =
1
2
3a?
3a = ( )3
6

30. a + b
-5
2( )a = b =
a+b
2
3
( )
a
b
-3
5= ( )
Calculate a + b, then represent the addition
diagrammatically.
Key Question.