# calc lecture01

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• MA 100 Mathematical Methods

Calculus Lecture 1

Introduction

Vectors and Lines

Department of Mathematics

London School of Economics and Political Science

• What is Calculus ?

from Wikipedia :

Calculus ( Latin, calculus, a small stone used for counting )is a branch in mathematics focused on limits, functions,

derivatives, integrals, and infinite series.

[. . . ]

Calculus is the study of change, in the same way that

geometry is the study of shape and algebra is the study of

operations and their application to solving equations.

[. . . ]

Calculus has widespread applications in science,

economics, and engineering and can solve many problems

for which algebra alone is insufficient.

MA 100, Mathematical Methods Calculus Lecture 1 page 2 / 31

• Real numbers

The real numbers are denoted by the symbol IR .

We often think as a real number as a point on a line, called

the real line ,

but we can also think of real numbers as displacementsalong the real line.

E.g., the number 2 also represents

a displacement of 2 units to the right.

MA 100, Mathematical Methods Calculus Lecture 1 page 3 / 31

• Real numbers

number displacement

-

7/3 1 0 1 2

MA 100, Mathematical Methods Calculus Lecture 1 page 4 / 31

• Vectors and the plane IR 2

The two-dimensional x , y -plane consists of vectors(

xy

)

. ( Note : we write vectors as columns. )

(

xy

)

has two interpretations :

a point in the plane :

position : x units in x -direction, y units in y -direction,

a displacement :

x units in x -direction and y in y -direction.

x and y are the components or coordinates of the vector.

MA 100, Mathematical Methods Calculus Lecture 1 page 5 / 31

• Vectors in the plane

point displacement

-

6

1 1 2

1

1

x -axis

y -axis

MA 100, Mathematical Methods Calculus Lecture 1 page 6 / 31

• Vectors

Vectors are often written in bold, v , or underlined, v ,to emphasise that theyre not numbers.

Vectors can be added and multiplied by scalars( a scalar is just a real number ).

Each operation can be interpreted algebraically andgeometrically.

MA 100, Mathematical Methods Calculus Lecture 1 page 7 / 31

• Operations on vectors

algebraically :

For vectors v =(

v1v2

)

and w =(

w1w2

)

, and IR :

v + w =(

v1 + w1v2 + w2

)

,

v =(

v1 v2

)

.

geometrically : . . .

MA 100, Mathematical Methods Calculus Lecture 1 page 8 / 31

• The sum of two vectors

(

22

)

+

(

31

)

=

(

51

)

=

(

31

)

+

(

22

)

-

6

1 1 2

1

1

x -axis

y -axis

MA 100, Mathematical Methods Calculus Lecture 1 page 9 / 31

• Product of scalar and vector

2(

31

)

=

(

62

)

-

6

1 1 2

1

1

x -axis

y -axis

MA 100, Mathematical Methods Calculus Lecture 1 page 10 / 31

• Length of a vector

u

-

6

*

v1

v2

MA 100, Mathematical Methods Calculus Lecture 1 page 11 / 31

• Length of a vector

The length of vector v =(

v1v2

)

satisfies

2 = v21 + v22

( Pythagoras Theorem ),

so the length, denoted v , is

v =

v21 + v22 .

MA 100, Mathematical Methods Calculus Lecture 1 page 12 / 31

• Distance between two vectors

u

u

-

6

:

HHHH

HHHH

HHY

w

vc

v = w + c , so c = v w

and hence the distance is c = v w .

MA 100, Mathematical Methods Calculus Lecture 1 page 13 / 31

• Distance between two vectors

The distance between two vectors v and w is

v w =

(v1 w1)2 + (v2 w2)2 .

MA 100, Mathematical Methods Calculus Lecture 1 page 14 / 31

• Scalar product of two vectors

The scalar product ( or inner product ) takes two vectorsand operates on them to give a real number ( i.e., a scalar ):

v , w =(

v1v2

)

,

(

w1w2

)

= v1 w1 + v2 w2 .

Notice : v , v = v21 + v22 = v

2 .

The scalar product looks algebraic,

but has important geometrical meanings.

MA 100, Mathematical Methods Calculus Lecture 1 page 15 / 31

• Algebraic properties of the scalar product

v , w = w , v

If IR , thenv ,w = v ,w , and v , w = v ,w ,

u + v ,w = u ,w + v ,w andu , v + w = u , v + u ,w .

Other properties follow,

such as u , v w = u , v u , w

etc.

MA 100, Mathematical Methods Calculus Lecture 1 page 16 / 31

• The cosine rule

u

u

-

6

:

HHHH

HHHH

HHY

w

vc

by Cosine Rule :

c2 = v2 + w2 2 v w cos

and by definition : c2 = v w2 ,

MA 100, Mathematical Methods Calculus Lecture 1 page 17 / 31

• More on the scalar product

so : v w2 = v2 + w2 2 v w cos

where is the angle between v and w .

Also : v w2 = v w , v w

= v , v w w , v w

= v , v v ,w w , v + w , w

= v2 + w2 2 v , w

and so : v , w = v w cos .

MA 100, Mathematical Methods Calculus Lecture 1 page 18 / 31

• Orthogonal vectors

Two non-zero vectors v and w are orthogonal orperpendicular or normal if the angle between them is /2 .

Since cos(

2

)

= 0, v and w are orthogonal precisely whenv ,w = 0.Example

Are(

24

)

and(

21

)

orthogonal ?

(

24

)

,

(

21

)

=

MA 100, Mathematical Methods Calculus Lecture 1 page 19 / 31

• 3-Dimensional space

3-dimensional space is denoted by IR3 .

Points / displacements are 3-dimensional vectors

(v1v2v3

)

.

Scalar product : v , w = v1 w1 + v2 w2 + v3 w3

Length : v =

v21 + v22 + v

23

etc. . . .

MA 100, Mathematical Methods Calculus Lecture 1 page 20 / 31

• Lines ( in 2-D first )

-

6

4

3

How do we describe the red line ?

MA 100, Mathematical Methods Calculus Lecture 1 page 21 / 31

• Lines

One way is to note that the points on the line are all obtained

from the vector(

30

)

by adding any scalar multiple of(

34

)

to it,

that is, each point x on the line satisfies

x =(

30

)

+ t(

34

)

, ( t IR ).

This is a Parametric Equation of the line.

MA 100, Mathematical Methods Calculus Lecture 1 page 22 / 31

• Lines in 3-D

Same story in IR3 :

x = + t v , ( t IR )

is the equation of the line through in the direction v .

In terms of components :

(xyz

)

=

(123

)

+ t

(v1v2v3

)

.

MA 100, Mathematical Methods Calculus Lecture 1 page 23 / 31

• Lines in 3-D

We can write this as :

(x 1y 2z 3

)

= t

(v1v2v3

)

,

and working out t gives :

t =x 1

v1=

y 2v2

=z 3

v3,

provided no v i is zero.

These are known as the Cartesian Equation(s) of the line.

MA 100, Mathematical Methods Calculus Lecture 1 page 24 / 31

• Lines in 3-D

Example

The line through =

( 101

)

in direction v =

(132

)

has

Cartesian equations

which simplifies to

MA 100, Mathematical Methods Calculus Lecture 1 page 25 / 31

• Lines in 2-D

The same works in 2 dimensions as well.

Example

The line(

xy

)

=

(

20

)

+ t(

11

)

has Cartesian equations

which simplifies to

MA 100, Mathematical Methods Calculus Lecture 1 page 26 / 31

• Lines in 2-D

Example

Is the point(

32

)

on the line(

xy

)

=

(

20

)

+ t(

11

)

?

If so, then we must have(

32

)

=

(

20

)

+ t(

11

)

, for some t .

That gives the equations

MA 100, Mathematical Methods Calculus Lecture 1 page 27 / 31

• Lines in 2-D

Same example

Is the point(

32

)

on the line(

xy

)

=

(

20

)

+ t(

11

)

?

Alternatively, the Cartesian equation of this line is

and(

xy

)

=

(

32

)

does

MA 100, Mathematical Methods Calculus Lecture 1 page 28 / 31

• Back to lines in 3-D

Example

Do the lines 1 :

(xyz

)

=

(101

)

+ t

(111

)

and 2 :

(xyz

)

= t

(201

)

intersect ?

If they do, then

MA 100, Mathematical Methods Calculus Lecture 1 page 29 / 31

• Back to lines in 3-D

That gives the system of equations

MA 100, Mathematical Methods Calculus Lecture 1 page 30 / 31

• Coplanar and skew in 3 dimensions

Two lines in 3-dimensional space are coplanar ( = lie inthe same plane ) if they are parallel or intersecting.