vectors intro

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Block 3 Vectors Introduction

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Page 1: Vectors intro

Block 3

Vectors Introduction

Page 2: Vectors intro

What is to be learned?

• What a vector is• Some other bits and pieces about vectors

Page 3: Vectors intro

A vector is

Like a journey

Need DistanceDirection

- magnitude

Page 4: Vectors intro

A

B

A

4

2

42( )B BA

4–

2–

-4 -2( ) components

Page 5: Vectors intro

A

B

– 3

AB

4

-3 4

( )

u˜ = =

Page 6: Vectors intro

C

D

-2

DC

4

4 -2

( )

t

= =

Page 7: Vectors intro

3 4( )u =

˜U = (3 , 4)

x

y U

components and coordinates

Page 8: Vectors intro

A

B

– 4

AB

2

-4 2

( )

u = =

Page 9: Vectors intro

Magnitude of a Vector

Page 10: Vectors intro

5

-4

5 -4( )

v =˜

magnitude of v =˜

√(52 +(-4)2))

Page 11: Vectors intro

5

-4

5 -4( )

v =˜

v =˜

√(52 +(-4)2))| |

Page 12: Vectors intro

3

-6

-6 3

( )

t

t˜ =

=t ?| | t˜| |

√((-6)2 + 32)

= √45

Page 13: Vectors intro

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

Page 14: Vectors intro

A

B

4

-3

4 -3( )AB =

= u˜

| u2 | =˜

42 + (-3)2

| u |˜

= 5

Page 15: Vectors intro

C

D

2

DC

-5

-5 2

( )

t

= = =t˜| |

√((-5)2 + 22)

= √29

Name the vector in two ways, write itscomponents and find its magnitude.

Key Question.

Page 16: Vectors intro

What is to be learned?

• How to add and “subtract” vectors.• How to multiply a vector by a scalar.

Page 17: Vectors intro

Vector Addition

Page 18: Vectors intro

a + b42( )a = b =

a+b

nose to tail

1 3

( )

a

b

5 5

= ( )

Page 19: Vectors intro

a + b ?( )a = b =( )

= ( )

4 5

2 3

( )4 5

( )2 3+

6 8

4 + 2

5 + 3

using components

laughably easy

Page 20: Vectors intro

Vector Subtraction

Page 21: Vectors intro

vector subtraction does not exist!(or does it?)

9 – 7= 9 + (-7)

we add the negative of the vector

Page 22: Vectors intro

5

5 -4( )

v =˜

- v =˜

4

-5 4

( )

Page 23: Vectors intro

a – b ?( )a = b =( )

= ( )

4 6

2 -3

-b =( )-2 3

a + -b

( )4 6

( )-2 3

+

2 9

4 – 2

6 – (-3)

Page 24: Vectors intro

Multiplying by a scalar

Page 25: Vectors intro

2 -1( )a = 3a

a

= a + a + a

a

a

3a = 6 -3( )

3a

Page 26: Vectors intro

Addition, Subtraction and Multiplication (by a Scalar)

Normal Number

Page 27: Vectors intro

a + b52( )a = b =

a+b

nose to tail

1 3

( )

a

b

6 5

= ( )adding

Page 28: Vectors intro

a – b ?( )a = b =( )= ( )

7 6

4 1

3 5

subtracting Theory – Add the negativePractice -

Page 29: Vectors intro

Scalar Multiplication

( )a = 1 2

3a?

3a = ( )3 6

Page 30: Vectors intro

a + b-5 2( )a = b =

a+b

2 3

( )

a

b

-3 5 = ( )

Calculate a + b, then represent the additiondiagrammatically.Key Question.