MA100 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 innite 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 insufcient.
MA100, 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 displacements
along the real line.
E.g., the number 2 also represents
a displacement of 2 units to the right.
MA100, Mathematical Methods Calculus Lecture 1 page 3/ 31
Real numbers
number displacement
-
7/3 1
0 1 2

MA100, Mathematical Methods Calculus Lecture 1 page 4/ 31
Vectors and the plane IR
2
The two-dimensional x, y -plane consists of vectors
_
x
y
_
. ( Note : we write vectors as columns. )
_
x
y
_
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.
MA100, Mathematical Methods Calculus Lecture 1 page 5/ 31
Vectors in the plane
point displacement
-
6
1 1 2
1
1
x -axis
y -axis
MA100, 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 and
geometrically.
MA100, Mathematical Methods Calculus Lecture 1 page 7/ 31
Operations on vectors
algebraically:
For vectors v =
_
v
1
v
2
_
and w =
_
w
1
w
2
_
, and IR:
v + w =
_
v
1
+ w
1
v
2
+ w
2
_
,
v =
_
v
1
v
2
_
.
geometrically: . . .
MA100, Mathematical Methods Calculus Lecture 1 page 8/ 31
The sum of two vectors
_
2
2
_
+
_
3
1
_
=
_
5
1
_
=
_
3
1
_
+
_
2
2
_
-
6
1 1 2
1
1
x -axis
y -axis
MA100, Mathematical Methods Calculus Lecture 1 page 9/ 31
Product of scalar and vector
2
_
3
1
_
=
_
6
2
_
-
6
1 1 2
1
1
x -axis
y -axis
MA100, Mathematical Methods Calculus Lecture 1 page 10/ 31
Length of a vector
u
-
6

*
v
1
v
2
MA100, Mathematical Methods Calculus Lecture 1 page 11/ 31
Length of a vector
The length of vector v =
_
v
1
v
2
_
satises

2
= v
2
1
+ v
2
2
( Pythagoras Theorem),
so the length, denoted v, is
v =
_
v
2
1
+ v
2
2
.
MA100, Mathematical Methods Calculus Lecture 1 page 12/ 31
Distance between two vectors
u
u
-
6

H
H
H
H
H
H
H
H
H
HY
w
v
c
v = w + c, so c = v w
and hence the distance is c = v w.
MA100, Mathematical Methods Calculus Lecture 1 page 13/ 31
Distance between two vectors
The distance between two vectors v and w is
v w =
_
(v
1
w
1
)
2
+ (v
2
w
2
)
2
.
MA100, Mathematical Methods Calculus Lecture 1 page 14/ 31
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 =
__
v
1
v
2
_
,
_
w
1
w
2
__
= v
1
w
1
+ v
2
w
2
.
Notice : v, v = v
2
1
+ v
2
2
= v
2
.
The scalar product looks algebraic,
but has important geometrical meanings.
MA100, Mathematical Methods Calculus Lecture 1 page 15/ 31
Algebraic properties of the scalar product
v, w = w, v
If IR, then
v, w = v, w, and v, w = 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.
MA100, Mathematical Methods Calculus Lecture 1 page 16/ 31
The cosine rule
u
u
-
6

H
H
H
H
H
H
H
H
H
HY
w
v
c

by Cosine Rule :
c
2
= v
2
+w
2
2 v w cos
and by denition : c
2
= v w
2
,
MA100, Mathematical Methods Calculus Lecture 1 page 17/ 31
More on the scalar product
so : v w
2
= v
2
+w
2
2 v w cos
where 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 .
MA100, Mathematical Methods Calculus Lecture 1 page 18/ 31
Orthogonal vectors
Two non-zero vectors v and w are orthogonal or
perpendicular or normal if the angle between them is /2.
Since cos
_

2
_
= 0, v and w are orthogonal precisely when
v, w = 0.
Example
Are
_
2
4
_
and
_
2
1
_
orthogonal ?
__
2
4
_
,
_
2
1
__
=
MA100, Mathematical Methods Calculus Lecture 1 page 19/ 31
3-Dimensional space
3-dimensional space is denoted by IR
3
.
Points / displacements are 3-dimensional vectors
_
v
1
v
2
v
3
_
.
Scalar product : v, w = v
1
w
1
+ v
2
w
2
+ v
3
w
3
Length : v =
_
v
2
1
+ v
2
2
+ v
2
3
etc. . . .
MA100, Mathematical Methods Calculus Lecture 1 page 20/ 31
Lines ( in 2-D rst )
-
6

4
3
How do we describe the red line ?
MA100, 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
_
3
0
_
by adding any scalar multiple of
_
3
4
_
to it,
that is, each point x on the line satises
x =
_
3
0
_
+ t
_
3
4
_
, ( t IR).
This is a Parametric Equation of the line.
MA100, Mathematical Methods Calculus Lecture 1 page 22/ 31
Lines in 3-D
Same story in IR
3
:
x = + t v, ( t IR)
is the equation of the line through in the direction v .
In terms of components :
_
x
y
z
_
=
_

3
_
+ t
_
v
1
v
2
v
3
_
.
MA100, Mathematical Methods Calculus Lecture 1 page 23/ 31
Lines in 3-D
We can write this as :
_
x
1
y
2
z
3
_
= t
_
v
1
v
2
v
3
_
,
and working out t gives :
t =
x
1
v
1
=
y
2
v
2
=
z
3
v
3
,
provided no v
i
is zero.
These are known as the Cartesian Equation(s) of the line.
MA100, Mathematical Methods Calculus Lecture 1 page 24/ 31
Lines in 3-D
Example
The line through =
_
1
0
1
_
in direction v =
_
1
3
2
_
has
Cartesian equations
which simplies to
MA100, Mathematical Methods Calculus Lecture 1 page 25/ 31
Lines in 2-D
The same works in 2 dimensions as well.
Example
The line
_
x
y
_
=
_
2
0
_
+ t
_
1
1
_
has Cartesian equations
which simplies to
MA100, Mathematical Methods Calculus Lecture 1 page 26/ 31
Lines in 2-D
Example
Is the point
_
3
2
_
on the line
_
x
y
_
=
_
2
0
_
+ t
_
1
1
_
?
If so, then we must have
_
3
2
_
=
_
2
0
_
+ t
_
1
1
_
, for some t .
That gives the equations
MA100, Mathematical Methods Calculus Lecture 1 page 27/ 31
Lines in 2-D
Same example
Is the point
_
3
2
_
on the line
_
x
y
_
=
_
2
0
_
+ t
_
1
1
_
?
Alternatively, the Cartesian equation of this line is
and
_
x
y
_
=
_
3
2
_
does
MA100, Mathematical Methods Calculus Lecture 1 page 28/ 31
Back to lines in 3-D
Example
Do the lines
1
:
_
x
y
z
_
=
_
1
0
1
_
+ t
_
1
1
1
_
and
2
:
_
x
y
z
_
= t
_
2
0
1
_
intersect ?
If they do, then
MA100, Mathematical Methods Calculus Lecture 1 page 29/ 31
Back to lines in 3-D
That gives the system of equations
MA100, Mathematical Methods Calculus Lecture 1 page 30/ 31
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 =
_
1
0
1
_
+ t
_
2
0
1
_
and x =
_
0
1
6
_
+ t
_
4
0
2
_
are parallel, hence coplanar.
The lines x =
_
1
0
1
_
+ t
_
1
1
1
_
and x = t
_
2
0
1
_
are
neither parallel nor intersecting;
such pairs of lines are called skew.
MA100, Mathematical Methods Calculus Lecture 1 page 31/ 31