Biomedical Engineering Reference
In-Depth Information
Substituting the second of (A.71) into the second equality of (A.72) one obtains
v
i
e
i
¼
Q
i
a
v
a
e
i
;
(A.73)
which may be rewritten as
ð
v
i
Q
i
a
v
a
Þ
e
i
¼
:
0
(A.74)
Taking the dot product of (A.74) with
e
j
, it follows that the sum over
i
is only
nonzero when
i
¼
j
, thus
v
j
¼
Q
j
a
v
a
:
(A.75)
If the first, rather than the second, of (A.71) is substituted into the second
equality of (A.72), and similar algebraic manipulations accomplished, one obtains
v
j
¼
Q
j
a
v
a
:
(A.76)
The results (A.75) and (A.76) are written in the matrix notation using
superscripted (L) and (G) to distinguish between components referred to the Latin
or the Greek bases:
v
ð
L
Þ
¼
v
ð
G
Þ
;
v
ð
G
Þ
¼
Q
T
v
ð
L
Þ
:
Q
(A.77)
Problems
A.6.1.
Is the matrix
2
3
212
4
5
1
3
1
22
2
2
1
an orthogonal matrix?.
A.6.2. Are the matrices
A
,
B
,
C
, and
Q
, where
Q
¼
C
B
A
, and where
2
4
3
5
;
2
4
3
5
;
2
4
3
5
cos
F
sin
F
0
1
0
0
cos
C
0
sin
C
010
sin
A
¼
sin
F
cos
F
0
B
¼
0
cos
y
sin
y
C
¼
0
0
1
0
sin
y
cos
y
C
0 s
C
all orthogonal matrices?
A.6.3. Does an inverse of the compositional transformation constructed in prob-
lem A.5.7 exist?
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