Biomedical Engineering Reference
In-Depth Information
In this case
A
is called the matrix representation of the second-order tensor
A
,as
the comparison of Eqs. (
1.36
) and (
1.39
) reveals.
Exercises
1.1
The basis
{
e
x
,
e
y
,
e
z
}
has a right-handed orientation and is orthonormal.
(a) Determine
|
e
i
|
for
i
=
x
,
y
,
z
.
(b) Determine
e
i
·
e
j
for
i
,
j
=
x
,
y
,
z
.
(c) Determine
e
x
·
e
y
×
e
z
.
(d) Why is:
e
x
×
e
y
=
e
z
?
Let
{
e
x
,
e
y
,
e
z
}
be an orthonormal vector basis. The force vectors
F
x
=
3
e
x
+
2
e
y
+
e
z
and
F
y
=−
4
e
x
+
e
y
+
4
e
z
act on point P. Calculate a
vector
F
z
acting on P in such a way that the sum of all force vectors is the
zero vector.
1.2
1.3
Let
{
e
x
,
e
y
,
e
z
}
be a right-handed and orthonormal vector basis. The follow-
ing vectors are given:
a
=
4
e
z
,
b
=−
3
e
y
+
4
e
z
and
c
=
e
x
+
2
e
z
.
(a) Write the vectors in column notation.
(b) Determine
a
+
b
and 3(
a
+
b
+
c
).
a
·
b
,
b
·
a
,
a
×
b
and
b
×
a
.
(c) Determine
|
b
×
b
|
b
(d) Determine
|
a
|
,
|
,
|
a
|
and
×
a
|
.
a
and
b
.
(f) Determine a unit normal vector on the plane defined by
(e) Determine the smallest angle between
a
and
b
.
×
b
·
b
.
(g) Determine
a
·
c
and
a
×
c
ab
ab
)
T
c
and
b
(h) Determine
·
c
,(
·
a
·
c
.
a
,
b
and
(i) Do the vectors
c
form a suitable vector basis? If the answer
is yes, do they form an orthogonal basis? If the answer is yes, do they
form an orthonormal basis?
Consider the basis
{
a
,
b
,
c
}
with
a
,
b
and
c
defined as in the previous
exercise. The following vectors are given:
d
=
a
+
2
b
and
e
=
2
a
−
3
c
.
(a) Determine
d
+
e
.
(b) Determine
d
·
e
.
1.4
1.5
The basis
{
e
x
,
e
y
,
e
z
}
is right-handed and orthonormal. The vectors
a
x
,
a
y
and
a
z
are given by:
a
x
=
4
e
x
+
3
e
y
;
a
y
=
3
e
x
−
4
e
y
and
a
z
=
a
x
×
a
y
.
(a) Determine
a
z
expressed in
e
x
,
e
y
and
e
z
.
(b) Determine
|
a
i
|
for
i
=
x
,
y
,
z
.
(c) Determine the volume of the parallelepiped defined by
a
x
,
a
y
and
a
z
.
a
x
and
a
y
.
(d) Determine the angle between the lines of action of
(e) Determine the vector
α
x
from
a
i
=|
a
i
|
α
i
for
i
=
x
,
y
,
z
.Is
{
α
x
,
α
y
,
α
z
}
a right-handed, orthonormal vector basis?
