Biomedical Engineering Reference
In-Depth Information
y
F
F
O
y
r
θ
d
x
x
FIGURE 4.4
(Left) The dot, or scalar, product of vectors
F
and
d
is equivalent to the projection of
F
onto
d
.
(Right) The cross, or vector, product of vectors
r
and
F
is a vector that points along the axis of rotation, the
z
-axis
coming out of the page.
of a force about a point or axis is a measure of its tendency to cause rotation.
The cross, or vector, product of two vectors yields a new vector that points along the axis
of rotation. For example, Figure 4.4 (right) shows a vector
F
acting in the
The
moment
plane at a
distance from the body's coordinate center O. The vector
r
points from O to the line of
action of
F
. The cross product
r
x
-
y
direction along the body's
axis of rotation. If
F
and
r
are three-dimensional, thereby including
k
components, their
cross product will have additional components of rotation about the
F
is a vector that points in the
z
x
and
y
axes. The
moment
M
resulting from crossing
r
into
F
is written
M
¼
M
x
i
þ
M
y
j
þ
M
z
k
ð
4
:
13
Þ
axes, respectively.
Cross products may be taken by crossing each vector component term by term—for
example:
M
x
,
M
y
, and
M
z
cause rotation of the body about the
,
, and
where
x
y
z
A
B
¼
3
ð
2
Þ
i
i
þ
3
ð
3
Þ
i
j
þ
3
ð
10
Þ
i
k
þ
2
ð
2
Þ
j
i
þ
2
ð
3
Þ
j
j
þ
2
ð
10
Þ
j
k
þ
1
ð
2
Þ
k
i
þ
1
ð
3
Þ
k
j
þ
1
ð
10
Þ
k
k
The magnitude
sin y, where y is the angle between
A
and
B
. Consequently, for
an orthogonal coordinate system, the cross products of all like terms equal zero, and
i
j
A
B
j¼
AB
j
¼
k
,
j
k
¼
i
,
k
i
¼
j
,
i
k
¼
j
, and so on. The previous example yields
A
B
¼
9
k
30
j
þ
4
k
þ
20
i
2
j
3
i
¼
17
i
32
j
þ
13
k
lb ft
Note that the cross product is
A
.
Cross products of vectors are commonly computed using matrices. The previous example
not
commutative—in other words,
A
B
6¼
B
A
B
is given by the matrix
