Graphics Reference
In-Depth Information
If the force is represented by
f
, and the mechanism's direction by
v
, then the magnitude of the
force acting in the direction of the mechanism is
cos . And as the total work done is the
product of the force over the acting distance, this is represented by
f
f
v
cos , which is the
dot product.
Figure 2.16 shows how the dot product should be visualized, where one vector is projected
onto the other and the two lengths multiplied.
C
B
AB
AC
C
′
AC
′
θ
A
Figure 2.16.
The two vectors are
−
AB and
−
AC with a separating angle . The projection of
−
AC onto
−
AB is
−
AC
, which equals
−
AC cos . The dot product of
−
AB and
−
AC is therefore
−
AB
−
AC
=
−
AB
−
AC
−
AB
·
−
AC
=
cos
Let us show that
a
·
b
=
a
b
cos
=
x
a
x
b
+
y
a
y
b
+
z
a
z
b
We begin with
a
=
x
a
i
+
y
a
j
+
z
a
k
and
b
=
x
b
i
+
y
b
j
+
z
b
k
Therefore,
·
=
+
+
·
+
+
a
b
x
a
i
y
a
j
z
a
k
x
b
i
y
b
j
z
b
k
Expanding
a
·
b
=
x
a
x
b
i
·
i
+
y
a
y
b
j
·
j
+
z
a
z
b
k
·
k
+
x
a
y
b
i
·
j
+
x
a
z
b
i
·
k
+
y
a
x
b
j
·
i
+
y
a
z
b
j
·
k
+
z
a
x
b
k
·
i
+
z
a
y
b
k
·
j
