Graphics Reference
In-Depth Information
B
Q
v
q
D
A
O
v
p
C
P
(a)
(b)
Figure 3.1
(a) The (free) vector
v
is not anchored to a specific point and may therefore
describe a displacement from any point, specifically from point
A
to point
B
, or from point
C
to point
D
. (b) A position vector is a vector bound to the origin. Here, position vectors
p
and
q
specify the positions of points
P
and
Q
, respectively.
A vector
v
can be interpreted as the displacement from the origin to a specific
point
P
, effectively describing the position of
P
. In this interpretation
v
is, in a sense,
bound
to the origin
O
. A vector describing the location of a point is called a
position
vector
,or
bound vector
. A vector
v
can also be interpreted as the displacement from
an initial point
P
to an endpoint
Q
,
Q
v
. In this sense,
v
is
free
to be applied at
any point
P
. A vector representing an arbitrary displacement is called a
free vector
(or
just
vector
). If the origin is changed, bound vectors change, but free vectors stay the
same. Two free vectors are equal if they have the same direction and magnitude; that
is, if they are componentwise equal. However, two bound vectors are not equal —
even if the vectors are componentwise equal — if they are bound to different origins.
Although most arithmetic operations on free and bound vectors are the same, there
are some that differ. For example, a free vector — such as the normal vector of a
plane — transforms differently from a bound vector.
The existence of position vectors means there is a one-to-one relationship between
points and vectors. Frequently, a fixed origin is assumed to exist and the terms
point
and
vector
are therefore used interchangeably.
As an example, in Figure 3.1a, the free vector
v
may describe a displacement from
any point, specifically from
A
to
B
,orfrom
C
to
D
. In Figure 3.1b, the two bound
vectors
p
and
q
specify the positions of the points
P
and
Q
, respectively, and only
those positions.
The vector
v
from point
A
to point
B
is written
v
=
P
+
=
−
AB
(which is equivalent to
=
−
OB
−
−
OA
). Sometimes the arrow is omitted and
v
is written simply as
v
v
AB
.
A special case is the vector from a point
P
to
P
itself. This vector is called the zero
vector, and is denoted by
0
.
In the context of working with vectors, real numbers are usually referred to as
scalars
. Scalars are here denoted by lowercase letters set in italics (for example
a
,
b
, and
c
).
=
