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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.