moment about the line L is then the projection of the vector moment MB on this
line. If ˆeLis a unit vector along the line, then MB·ˆeLrepresents the projection of
MBon L. The direction of the unit vector ˆeLon the line Lcan point in one of two
directions (i.e. ˆeLor −ˆeL). However, once the direction of ˆeL has been chosen one
must be careful to analyze the dot product MB·eˆLas its algebraic sign determines
the rotation sense produced by the moment (i.e., clockwise or counterclockwise).
Aresultant force is the algebraic sum of the forces associated with a system.
The moment of a resultant force with respect to some axis is equal to the algebraic
sum of the moments of the system forces with respect to the same axis.
Example 6-16. If F =F 1 ˆe 1 +F 2 ˆe 2 +F 3 ˆe 3 is a force acting at the end of the
position vector r =xˆe 1 +yˆe 2 +zˆe 3 then the moment of the force about the origin is
M =r ×F = (yF 3 −zF 2 )ˆe 1 + (zF 1 −xF 3 )ˆe 2 + (xF 2 −yF 1 )ˆe 3. Make note that the moments
of the force components are
M 1 =r ×(F 1 ˆe 1 ) = (xˆe 1 +yˆe 2 +zˆe 3 )×(F 1 ˆe 1 ) = −yF 1 ˆe 3 +zF 1 ˆe 2
M 2 =r ×(F 2 ˆe 2 ) = (xˆe 1 +yˆe 2 +zˆe 3 )×(F 2 ˆe 2 ) = xF 2 ˆe 3 −zF 2 ˆe 1
M 3 =r ×(F 3 ˆe 3 ) = (xˆe 1 +yˆe 2 +zˆe 3 )×(F 3 ˆe 3 ) = −xF 3 ˆe 2 +yF 3 eˆ 1
so that M =M 1 +M 2 +M 3
Differentiation of Vectors
Let us define what is meant by a derivative associated with a vector and consider
some applications of these derivatives. Again notation plays an important part in
the representation of the derivatives and therefore many examples are given to help
clarify concepts as they arise.
The equation of a space curve can be described in terms of a position vector from
the origin of a chosen coordinate system. For example, in cartesian coordinates the
position vector of a space curve can have the form
r =r (t) = x(t)ˆe 1 +y(t)ˆe 2 +z(t)ˆe 3 , (6 .47)
where the space curve is defined by the parametric equations
x=x(t), y =y(t), z =z(t). (6 .48)
