Graphics Reference
In-Depth Information
Definition.
Two vectors
u
and
v
in an
arbitrary
vector space with inner product •
are said to be
orthogonal
if
u
•
v
= 0.
Let
u
,
v
Œ
R
n
.
1.3.1. Theorem.
(1)
u
^
v
if and only if
u
and
v
are orthogonal.
(2)
u
||
v
if and only if
u
and
v
are linearly dependent.
Proof.
Most of the theorem follows easily from the definitions. Use the Cauchy-
Schwarz inequality to prove (2).
Although the words “orthogonal” and “perpendicular” have different connota-
tions, Theorem 1.3.1 shows that they mean the same thing and we have an extremely
easy test for this property, namely, we only need to check that a dot product is zero.
Checking whether two vectors are parallel is slightly more complicated. We must
check if one is a multiple of the other.
Finally, note that if
u
= (u
1
,u
2
,...,u
n
) is a unit vector, then u
i
=
u
•
e
i
= cos q
i
, where
q
i
is the angle between
u
and
e
i
. This justifies the following terminology:
1
v
Definition.
If
v
is a nonzero vector, then the ith component of the unit vector
is called the
ith direction cosine
of
v
.
v
1.4
Inner Product Spaces: Orthonormal Bases
This section deals with some very important concepts associated with
arbitrary
vector
spaces with an inner product. We shall use the dot notation for the inner product. The
reader may, for the sake of concreteness, mentally replace every phrase “vector space”
with the phrase “vector subspace of
R
n
or
C
n
,” but should realize that everything we
do here holds in the general setting.
Probably the single most important aspect of inner product spaces is the existence
of a particularly nice type of basis.
Definition.
If
v
1
,
v
2
,...,
v
n
are vectors in an inner product space, we say that they
are
mutually orthogonal
if
v
i
•
v
j
= 0 for i π j. A set of vectors is said to be a
mutually
orthogonal set
if it is empty or its vectors are mutually orthogonal.
Definition.
Let
V
be an inner product space and let B be a basis for
V
. If B is a
mutually orthogonal set of vectors, then B is called an
orthogonal basis
for
V
. If, in
addition, the vectors of B are all unit vectors, then B is called an
orthonormal basis
.
In the special case where
V
consists of only the zero vector, it is convenient to call the
empty set an
orthonormal basis for
V
.
Orthonormal bases are often very useful because they can greatly simplify com-
putations. For example, if we wanted to express a vector
v
in terms of a basis
v
1
,
v
2
,
...,
v
n
, then we would normally have to solve the linear equations
vv v
=
a
+
a
2 2
...
+
+
a
nn
v
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