MAT1341A: LECTURE NOTES FOR FALL 2008
MONICA NEVINS (HEAVILY BORROWING FROM HANDWRITTEN NOTES OF BARRY JESSUP)
2.
Vector Geometry, Part I
The material for this class and the next comes largely from [N, 3.1,3.2,3.3,3.5], but we also
incorporate the generalizations to higher dimensions where appropriate. We will not cover
everything in these sections, and we will add some material that is not present in the book.
You are responsible for what we cover in class, the suggested exercises, what is on the
homework assignments, and what is presented in the DGD.
It is assumed that [N, 3.1,3.2,3.3] are substantially review for students having taken the
prerequisites for this course; but this review gives us the opportunity to set the stage for the
upcoming chapters.
You are encouraged to look through those sections of the textbook to refresh your memory.
One good quick technique is to consider all the words in
boldface
, and read more about
any of those keywords which are unfamiliar.
2.1.
Vectors in
R
n
[N, 3.1]
.
Vector
comes from the Latin
vehere
, which means to carry, or
to convey; abstractly we think of the vector as taking us along the arrow that we represent
it with. For example, we use vectors in Physics to indicate the magnitude and direction of
a force.
Let’s use our understanding of the geometry and algebra of vectors in low dimensions (2 and
3) to develop some ideas about the geometry and algebra in higher dimensions.
“Algebra”
“Geometry”
R
, real numbers, “scalars”
real line
R
2
=
{
(
x, y
)

x, y
∈
R
}
, vectors
~u
= (
x, y
)
plane
R
3
=
{
(
x, y, z
)

x, y, z
∈
R
}
, vectors
= (
x, y, z
)
3space
.
.
.
(why not keep going?)
.
.
.
R
4
=
{
(
x
1
, x
2
, x
3
, x
4
)

x
i
∈
R
}
, vectors
~x
= (
x
1
, x
2
, x
3
, x
4
)
Hamilton (1843): extended
C
to “hamiltonians”
“spacetime”
n
∈
Z
,
n >
0:
R
n
=
{
(
x
1
, x
2
,
···
, x
n
)

x
i
∈
R
}
= (
x
1
,
, x
n
)
“
n
space” [N, 4.1.2]
Date
: September 9, 2008.
1