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An Vectors 2:Algebra of Vectors

Department of MathematicsUniversity of Leicester

Contents

Introduction

Magnitude

Vector Addition

Scalar Multiplication

Introduction

• A vector has size and direction.

• The size or magnitude of a vector means the length from its start point to its end point.

• You can add vectors together, and also multiply them by scalars.

Introduction

Magnitude

Vector Addition

Scalar Multiplication

Next

x

yv

Magnitude = length

• Take the vector

• Then its magnitude is found by Pythagoras’s Theorem:

• If its magnitude is 1, it is a unit vector

Magnitude of a Vector

22|| bav

Introduction

Magnitude

Vector Addition

Scalar Multiplication

Click here to see a

proof

b

av

Next

x

y

x

y

22 ba b

a

Click here to repeat

Click here to go back(by Pythagoras’s Theorem)

Magnitude of a Vector – 3 Dimensions

Consider the vector (a, b, c) in 3D and it’s projection onto the x-y plane.

By Pythagoras’ Theorem

we know the magnitude

of this is

x

y

a

b

22 ba

Introduction

Magnitude

Vector Addition

Scalar Multiplication

Next

Now consider how far it goes up in the z direction

Magnitude of a Vector – 3 Dimensions

z

We know the length of

this is .22 ba

We know the length of this is c from the origin

Again, by Pythagoras:

22222

22 cbacba

Introduction

Magnitude

Vector Addition

Scalar Multiplication

x-y plane

Next

21

56

10

Questions…

What is the magnitude of ?

7

3

Introduction

Magnitude

Vector Addition

Scalar Multiplication

82 8

Questions…

What is the magnitude of this vector:

?x

y

2

4

20

Introduction

Magnitude

Vector Addition

Scalar Multiplication

Vector Addition

• To add vectors together, we add together the elements of the same rows

• Take the vectors and

7

4

2

5

3

3

2

1

5

5

3

3

7

4

2

Introduction

Magnitude

Vector Addition

Scalar Multiplication

Next

Vector Addition- Geometry

b

a

a+b

Click here to see how the vectors add

together

x

y

Introduction

Magnitude

Vector Addition

Scalar Multiplication

Next

x

x

y

y

x

y

b

a

x

y

aa+b

b

OR:

Create the parallelogram

Draw the 2nd vector

Draw the 1st vector

Draw the 1st vector

Draw the 2nd vector STARTING FROM

the first vector

The end is the new vector

Back to Vector

Addition

Repeat

Repeat

a+b

x

y

b

a

x

y

aa+b

b

OR:

Create the parallelogram

Draw the 2nd vector

Draw the 1st vector

Draw the 1st vector

Draw the 2nd vector STARTING FROM

the first vector

The end is the new vector

Back to Vector

Addition

Repeat

Repeat

a+b

x

y

b

a

x

y

aa+b

b

OR:

Create the parallelogram

Draw the 2nd vector

Draw the 1st vector

Draw the 1st vector

Draw the 2nd vector STARTING FROM

the first vector

The end is the new vector

Back to Vector

Addition

Repeat

Repeat

a+b

x

y

b

a

x

y

aa+b

b

OR:

Create the parallelogram

Draw the 2nd vector

Draw the 1st vector

Draw the 1st vector

Draw the 2nd vector STARTING FROM

the first vector

The end is the new vector

Back to Vector

Addition

Repeat

Repeat

a+b

x

y

b

a

x

y

aa+b

b

OR:

Create the parallelogram

Draw the 2nd vector

Draw the 1st vector

Draw the 1st vector

Draw the 2nd vector STARTING FROM

the first vector

The end is the new vector

Back to Vector

Addition

Repeat

Repeat

a+b

+ +

2

-2

4

6

8

-6

-8

-4

0-2-6 -4 2 4 6 8-8

Introduction

Magnitude

Vector Addition

Scalar Multiplication

Next

Blue are the vectors,Pink is the sum.

y

xDraw these added together

Clear

Draw this vector Draw this vectorDraw this vector

Show the vectors drawn top-to-tail

Scalar Multiplication

• To multiply by a scalar, we just multiply each part of the vector by the scalar, individually.

c

b

a

c

b

a

Introduction

Magnitude

Vector Addition

Scalar Multiplication

Next

Scalar Multiplication- Geometry

x

y

a

2a

(-1)a

Introduction

Magnitude

Vector Addition

Scalar Multiplication

Next

Try multiplying some vectors by scalars

2

-2

4

6

8

-6

-8

-4

0-2-6 -4 2 4 6 8-8 v is pink

λv blue

Introduction

Magnitude

Vector Addition

Scalar Multiplication

Next

y

x

v Draw vector

Clear

Draw scaled vector

Draw both together

37

9

3

9

7

Questions…

What is ?

6

5

3

2

Introduction

Magnitude

Vector Addition

Scalar Multiplication

24

84

Questions…

What is ?

21

4

2

5

Introduction

Magnitude

Vector Addition

Scalar Multiplication

Questions…

Which of these vectors is 3a + b?

x

y

x

y

a b

Introduction

Magnitude

Vector Addition

Scalar Multiplication

Conclusion

• The magnitude of a vector can be found using Pythagoras’s theorem.

• This can be extended to any number of dimensions.

• Vectors can be added and multiplied by scalars.

• (You can’t multiply two vectors together).

Introduction

Magnitude

Vector Addition

Scalar Multiplication

Next