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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 twodimensional 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 nonzero 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
3Dimensional space
3dimensional space is denoted by IR3 .
Points / displacements are 3dimensional 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 2D 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 3D
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 3D
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 3D
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 2D
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 2D
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 2D
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 3D
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 3D
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 3dimensional space are coplanar ( = lie inthe same plane ) if they are parallel or intersecting.