Biomedical Engineering Reference
In-Depth Information
consider the elementary definition of the scalar product below as the definition of
the angle
z
,
u
v
¼
u
i
v
i
¼j
u
jj
v
j
cos
z
(A.61)
z
as the angle between the
two vectors
u
and
v
in two or three dimensions, it may seem strange to have the cos
Recalling that there is a geometric interpretation of
z
appear in the formula (A.61), which is valid in
n
dimensions. However, since
u
v
divided by
jujjvj
is always less than one and thus the definition (A.61) is
reasonable not only for two and three dimensions, but for a space of any finite
dimension. It is only in two and three dimensions that the angle
z
may be interpreted
as the angle between the two vectors.
Example A.6.1
Show that the magnitude of the sum of the two unit vectors
e
1
¼
[1,0] and
e
2
¼
[cos
a
, sin
a
] can vary in magnitude from 0 to 2, depending on the value of the angle
a
.
Solution:
e
1
þ
e
2
¼
[1
þ
cos
a
, sin
a
], thus
e
1
þ
e
2
│
¼
√
2
(1
þ
cos
a
). It follows
│
√
that
e
1
þ
e
2
│
¼
2 when
a ¼
0,
e
1
þ
e
2
│
¼
0 when
a ¼ p
, and
e
1
þ
e
2
│
¼
√
2
│
│
│
when
/2. Thus the sum of two unit vectors in two dimensions can point in any
direction in the two dimensions and can have a magnitude between 0 and 2.
A set of unit vectors
e
i
,
i
a ¼ p
,
n
, is called an
orthonormal or cartesian basis
of the vector space if all the base vectors are of unit magnitude and are orthogonal to
each other,
e
i
¼
1, 2,
...
e
j
¼ d
ij
for
i
,
j
having the range
n
. From the definition of orthogonality
one can see that, when
i
6¼
j
, the unit vectors
e
i
and
e
j
are orthogonal. In the case
where
i
j
the restriction reduces to the requirement that the
e
i
's be unit vectors.
The elements of the n-tuples
v
¼
,
v
n
] referred to an orthonormal basis
are called cartesian components. An important question concerning vectors is the
manner in which their components change as their orthonormal basis is changed. In
order to distinguish between the components referred to two different bases of a
vector space we introduce two sets of indices. The first set of indices is composed of
the lowercase Latin letters
i
,
j
,
k
,
m
,
n
,
p
, etc. which have the admissible values 1, 2, 3,
...
¼
[
v
1
,
v
2
,
v
3
,
...
n
as before; the second set is composed of the lowercase Greek letters
a
,
b
,
g
,
d
,
...
,
n
. The
Latin basis refers to the base vectors
e
i
while the Greek basis refers to the base
vectors
e
a
. The components of a vector
v
referred to a Latin basis are then
v
i
,
i
etc. whose set of admissible values are the Roman numerals I, II, III,
...
¼
1, 2,
3,
...
,
n
, while the components of the same vector referred to a Greek basis are
v
a
,
a ¼
,
n
. It should be clear that
e
1
is not the same as
e
I
,
v
2
is not the same as
v
II
, etc., that
e
1
,
v
2
refer to the Latin basis while
e
I
, and
v
II
refers to the Greek basis.
The terminology of calling a set of indices “Latin” and the other “Greek” is arbitrary;
we could have introduced the second set of indices as
i
0
,
j
0
,
k
0
,
m
0
,
n
0
,
p
0
, etc., which
would have had admissible values of 1
0
,2
0
,3
0
,
I, II, III,
...
...
,
n
, and subsequently spoken of the
unprimed and primed sets of indices.
The range of the indices in the Greek and Latin sets must be the same since both
sets of base vectors
e
i
and
e
a
occupy the same space. It follows then that the two
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