Graphics Reference
In-Depth Information
2
v
v
-
v
(a)
(b)
(c)
Figure 3.3
(a) The vector
v
. (b) The negation of vector
v
. (c) The vector
v
scaled by a
factor of 2.
When the scalar
k
is negative,
w
has a direction opposite that of
v
. The
length
of a
vector
v
is denoted by
v
and is defined in terms of its components as
(
v
1
+
=
v
2
+···+
v
n
).
v
The length of a vector is also called its
norm
or
magnitude
. A vector with a magnitude
of 1 is called a
unit vector
. A nonzero vector
v
can be made unit, or be
normalized
,by
multiplying it with the scalar 1/
.
For any
n
-dimensional vector space (specifically
v
n
) there exists a basis consisting
of exactly
n
linearly independent vectors
e
1
,
e
2
,
...
,
e
n
. Any vector
v
in the space can
be written as a
linear combination
of these base vectors; thus,
R
v
=
a
1
e
1
+
a
2
e
2
+···+
a
n
e
n
,
where
a
1
,
a
2
,
...
,
a
n
are scalars.
Given a set of vectors, the vectors are
linearly independent
if no vector of the set can
be expressed as a linear combination of the other vectors. For example, given two
linearly independent vectors
e
1
and
e
2
, any vector
v
in the plane spanned by these
two vectors can be expressed linearly in terms of the vectors as
v
=
a
1
e
1
+
a
2
e
2
for
some constants
a
1
and
a
2
.
Most bases used are
orthonormal
. That is, the vectors are pairwise orthogonal and
are unit vectors. The standard basis in
3
is orthonormal, consisting of the vectors
(1, 0, 0), (0, 1, 0), and (0, 0, 1), usually denoted, respectively, by
i
,
j
, and
k
.
R
3.3.2
Algebraic Identities Involving Vectors
Given vectors
u
,
v
, and
w
, the following identities hold for vector addition and
subtraction:
u
+
v
=
v
+
u
+
+
=
+
+
(
u
v
)
w
u
(
v
w
)
