Graphics Reference
In-Depth Information
t
xq
+= +.
xq
t
xq
xq
In other words,
t
+= +
1
t
1
(1.9)
It is easy to show that the only solutions to (1.9) are 0 £ t. But the equation
px
= t
xq
can be rewritten as
t
xp
=+
+
pq
1
t
which shows that
x
Π[
p
,
q
] since 0 £ t/(1 + t) £ 1.
The next proposition proves another fairly innocuous looking fact. It also plays a
key role in the proofs of a number of future theorems.
1.2.4. Proposition.
Let
p
be a point on a line
L
. If c > 0, then there are two and
only two points
x
on
L
that satisfy the equation |
px
| = c.
Proof.
Let
q
be a point on
L
distinct from
p
. Then any point
x
on
L
has the form
x
=
p
+ s
pq
and hence c = |
px
| = |s| |
pq
|. The only solutions to |s| = c/|
pq
| are s =±t,
where t = c/|
pq
|. In other words,
xp pq
=+
t
r
xp pq
=-
t
and the proposition is proved.
Finally,
Definition.
Let
p
,
v
,
q
Œ
R
n
. If
v
π
0
, then the
ray from
p
in direction
v
, denoted by
ray(
p
,
v
), is defined by
(
)
=+
{
0.
ray
pv
,
p
t
v
£
t
If
p
π
q
, then the
ray from
p
through
q
, denoted by [
pq
>, is defined by
[
(
)
pq
>
=
ray
p pq
,
.
1.3
Angles
The intuitive notion of the angle between two vectors is something that one picks up
early, probably while studying Euclidean geometry in high school. In this section we
show that there is a very simple rigorous definition of this that is also very easy to
compute. Everything we do here holds for an arbitrary real vector space with an inner
product, but, for the sake of concreteness, we restrict the discussion to Euclidean
space with its standard dot product.
