Calc Lecture01

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.


Real numbers

number ←→ displacement

-

−7/3 −1 0 1 2 π


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.


Vectors in the plane

point ←→ displacement

-

6

−1 1 2

−1

1

x -axis

y -axis


Vectors

Vectors are often written in bold, v , or underlined, v ,

to emphasise that they’re not numbers.

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

Each operation can be interpreted ‘algebraically’ and‘geometrically’.


Operations on vectors

algebraically :

For vectors v =

(

v1

v2

)

and w =

(

w1

w2

)

, and α ∈ IR :

v + w =

(

v1 + w1

v2 + w2

)

,

α v =

(

α v1

α v2

)

.

geometrically : . . .


The sum of two vectors

(

22

)

+

(

3−1

)

=

(

51

)

=

(

3−1

)

+

(

22

)

-

6

−1 1 2

−1

1

x -axis

y -axis


Product of scalar and vector

2(

31

)

=

(

62

)

-

6

−1 1 2

−1

1

x -axis

y -axis


Length of a vector

u

-

6

��*

v1

v2


Length of a vector

The length ℓ of vector v =

(

v1

v2

)

satisfies

ℓ2= v2

1 + v22

( Pythagoras’ Theorem ),

so the length, denoted ‖v‖ , is

‖v‖=√

v21 + v2

2 .


Distance between two vectors

u

u

-

6

��:

��

HHHHHHHHHHY

w

vc

v = w + c , so c = v − w

and hence the distance is ‖c‖ = ‖v − w‖ .


Distance between two vectors

The distance between two vectors v and w is

‖v − w‖=√

(v1 − w1)2 + (v2 − w2)2 .


Scalar product of two vectors

The scalar product ( or inner product ) takes two vectors

and operates on them to give a real number ( i.e., a scalar ):

〈v , w 〉=⟨(

v1

v2

)

,

(

w1

w2

)⟩

= v1 w1 + v2 w2 .

Notice : 〈v , v 〉 = v21 + v2

2 = ‖v‖2 .

The scalar product looks ‘algebraic’,

but has important geometrical meanings.


Algebraic properties of the scalar product

〈v , w 〉 = 〈w , v 〉

If α ∈ IR , then

〈αv , w 〉 = α 〈v , w 〉 , and 〈v , αw 〉 = α 〈v , w 〉 ,

〈u + v , w 〉 = 〈u , w 〉+ 〈v , w 〉 and

〈u , v + w 〉 = 〈u , v 〉+ 〈u , w 〉 .

Other properties follow,

such as 〈u , v − w 〉 = 〈u , v 〉 − 〈u , w 〉

etc.


The cosine rule

u

u

-

6

��:

��

HHHHHHHHHHY

w

vc

θ

by Cosine Rule :

‖c‖2 = ‖v‖2 + ‖w‖2 − 2 ‖v‖ ‖w‖ cos θ

and by definition : ‖c‖2 = ‖v − w‖2 ,


More on the scalar product

so : ‖v − w‖2 = ‖v‖2 + ‖w‖2 − 2 ‖v‖ ‖w‖ cos θ

where θθθθθθθθθ is the angle between v and w .

Also : ‖v − w‖2 = 〈v − w , v − w 〉

= 〈v , v − w 〉 − 〈w , v − w 〉

= 〈v , v 〉 − 〈v , w 〉 − 〈w , v 〉+ 〈w , w 〉

= ‖v‖2 + ‖w‖2 − 2 〈v , w 〉

and so : 〈v , w 〉 = ‖v‖ ‖w‖ cos θ .


Orthogonal vectors

Two non-zero vectors v and w are orthogonal or

perpendicular or normal if the angle between them is π/2 .

Since cos(

π

2

)

= 0, v and w are orthogonal precisely when〈v , w 〉 = 0.Example

Are(

24

)

and(

2−1

)

orthogonal ?

⟨(

24

)

,

(

2−1

)⟩

=


3-Dimensional space

3-dimensional space is denoted by IR3 .

Points / displacements are 3-dimensional vectors

(v1

v2

v3

)

.

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

Length : ‖v‖ =√

v21 + v2

2 + v23

etc. . . .


Lines ( in 2-D first )

-

6

��

��

��

��

��

��

��

��

��

4

−3

How do we describe the red line ?


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.


Lines in 3-D

Same story in IR3 :

x = ξξξξξξξξξ + t v , ( t ∈ IR )

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

In terms of components :

(xyz

)

=

(ξ1

ξ2

ξ3

)

+ t

(v1

v2

v3

)

.


Lines in 3-D

We can write this as :

(x − ξ1

y − ξ2

z − ξ3

)

= t

(v1

v2

v3

)

,

and working out t gives :

t =x − ξ1

v1=

y − ξ2

v2=

z − ξ3

v3,

provided no v i is zero.

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


Lines in 3-D

Example

The line through ξξξξξξξξξ =

( 10−1

)

in direction v =

(−132

)

has

Cartesian equations

which simplifies to


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


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


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


Back to lines in 3-D

Example

Do the lines ℓ1 :

(xyz

)

=

(101

)

+ t

(−1−1−1

)

and ℓ2 :

(xyz

)

= t

(201

)

intersect ?

If they do, then


Back to lines in 3-D

That gives the system of equations


Coplanar and skew in 3 dimensions

Two lines in 3-dimensional space are coplanar ( = lie in

the same plane ) if they are parallel or intersecting.

Ex. : x =

(101

)

+ t

( 20−1

)

and x =

(016

)

+ t

(−402

)

are parallel, hence coplanar.

The lines x =

(101

)

+ t

(−1−1−1

)

and x = t

(201

)

are

neither parallel nor intersecting;

such pairs of lines are called skew .MA 100, Mathematical Methods – Calculus – Lecture 1 – page 31 / 31

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Calc Lecture01