UNIVERSITY OF NEW SOUTH WALES
SCHOOL OF MATHEMATICS AND STATISTICS
MATH 1131
MATHEMATICS 1A ALGEBRA.
Section 2:  Vector Geometry.
In Chapter 1, see looked at the rudiments of vector geometry. We now have enough machin
ery to study this subject in greater depth.
Distances and Lengths:
We saw earlier that:
in higher dimensions, we can define the length of a vector as fol
lows:
Definition:
A vector
x
=
a
1
a
2
.
.
.
a
n
in
R
n
has length

x

given by

x

=
p
a
2
1
+
a
2
2
+
· · ·
+
a
2
n
.
The distance between two points
A
and
B
in
R
n
will be defined as the length of the vec
tor
→
AB
, in other words, if
A
has position vector
a
=
a
1
a
2
.
.
.
a
n
and
B
has position vector
b
=
b
1
b
2
.
.
.
b
n
, then the length of
→
AB
is

→
AB

=

b

a

=
p
(
b
1

a
1
)
2
+
· · ·
+ (
b
n

a
n
)
2
.
A vector which has unit length is called a
unit vector
. Any nonzero vector can be made
into a unit vector by dividing by its length.
Dot Product:
We define the
dot product
of two vectors
a
=
a
1
a
2
a
3
and
b
=
b
1
b
2
b
3
by
a
·
b
=
a
1
b
1
+
a
2
b
2
+
a
3
b
3
.
Note that this is a scalar, and so the dot product is often called a
scalar product
.
Ex:
Find the dot product of
a
=
1
2
3
and
b
=

1
7

3
1