The total moment about the origin is a vector quantity represented as the vector
sum of the above moments in the form
M 0 =M 1 eˆ 1 +M 2 ˆe 2 +M 3 ˆe 3
= (F 3 y 1 −F 2 z 1 )ˆe 1 + (F 1 z 1 −F 3 x 1 )ˆe 2 + (F 2 x 1 −F 1 y 1 )ˆe 3.
(6 .45)
If r 1 =x 1 ˆe 1 +y 1 ˆe 2 +z 1 ˆe 3 is the position vector from the origin to the point (x 1 , y 1 , z 1 ),
then the moment about the origin produced by the force F can be expressed as a
cross product of the vectors r 1 and F and written as
M 0 =r 1 ×F=
∣∣
∣∣
∣∣
ˆe 1 eˆ 2 ˆe 3
x 1 y 1 z 1
F 1 F 2 F 3
∣∣
∣∣
∣∣. (6 .46)
This is readily verified by expanding the equation (6.46) and showing the result is
given by equation (6.45).
Moment About Arbitrary Line
Assume one has a force F acting through a
given point A and r A is the position from the
origin to the point A. The moment about the
origin is given by M 0 =r A×F. The moment of
the given force F about the lines representing the
x,yand zaxes are given by the projections of M 0
on each of these axes. One finds these moments
M 0 ·eˆ 1 =M 1 , M 0 ·ˆe 2 =M 2 , M 0 ·ˆe 3 =M 3
To find the moment about a given line L, choose any point B on the line L
and construct the position vector r B from the origin to the point B. The vector
r A−r Bthen points from point Bto the force F acting at point Aas illustrated in
the previous figure.
The moment of the force F about the point Bis given by
MB= (r A−r B)×F
Observe that this equation for MB represents a position vector from point B to
the force F crossed with F and has the exact same form as equation (6.46). The
only difference being where the position vector to the force F is constructed. The
