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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 π
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 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’.
MA 100, Mathematical Methods – Calculus – Lecture 1 – page 7 / 31
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 : . . .
MA 100, Mathematical Methods – Calculus – Lecture 1 – page 8 / 31
The sum of two vectors
(
22
)
+
(
3−1
)
=
(
51
)
=
(
3−1
)
+
(
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 ℓ of vector v =
(
v1
v2
)
satisfies
ℓ2= v2
1 + v22
( Pythagoras’ Theorem ),
so the length, denoted ‖v‖ , is
‖v‖=√
v21 + v2
2 .
MA 100, Mathematical Methods – Calculus – Lecture 1 – page 12 / 31
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‖ .
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 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.
MA 100, Mathematical Methods – Calculus – Lecture 1 – page 15 / 31
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.
MA 100, Mathematical Methods – Calculus – Lecture 1 – page 16 / 31
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 ,
MA 100, 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 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 .
MA 100, 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 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
)⟩
=
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
(v1
v2
v3
)
.
Scalar product : 〈v , w 〉 = v1 w1 + v2 w2 + v3 w3
Length : ‖v‖ =√
v21 + v2
2 + v23
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 = 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
)
.
MA 100, Mathematical Methods – Calculus – Lecture 1 – page 23 / 31
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.
MA 100, Mathematical Methods – Calculus – Lecture 1 – page 24 / 31
Lines in 3-D
Example
The line through ξξξξξξξξξ =
( 10−1
)
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 ℓ1 :
(xyz
)
=
(101
)
+ t
(−1−1−1
)
and ℓ2 :
(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 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