STATICS  THEORY



The Cross Product

Cross Product Direction and Sign


The dot
product
was previously introduced as a way of "multiplying"
vectors where the result is a scalar,
A • B = AB cosθ
The second method for "multiplying" vectors is called the cross
product, and the result is a vector. The cross product of two vectors
A and B gives the vector C that has a magnitude of
C = A ×
B = AB sinθ
Here θ is the angle between the vectors A
and B.
The vector C is perpendicular to the plane defined by the vectors
A and B, and the direction is determined from the righthand
rule. 





Cross Product with Cartesian Vectors



If the vectors A and B are in Cartesian form, then it
can be shown that the cross product of A and B is given
by the determinant of the unit vectors and the Cartesian components
of A
and B.
It is interesting to note that this determinant could also be written as,
The results will be the same for either form.




Vectors r and F are
always
perpendicular to M ( = r × F)
Cross Product Direction and Sign


Moment as a Cross Product


If the moment of a force F about the point O is represented by
the vector M_{o}, then it can be shown that
M_{o} = r ×
F
Here r is the position vector of any point on the line of action
of F with respect to point O. Expanding the determinant form
of the cross product, gives
M_{o} = (r_{y} F_{z}
 r_{z} F_{y})i + (r_{z} F_{x}
 r_{x} F_{z})j
+ (r_{x} F_{y}  r_{y} F_{x})k
From this equation, the Cartesian components of the moment vector M_{o}
are readily found.






Moment of a Force About a Line or Axis

Moment About an Axis 

In many problems, it is necessary to determine the moment of a force
about a certain line or axis, such as an axle on a car. Since the moment
of a force is a vector, the dot product can be used to determine its
component parallel to any line a.
M_{a} = (M_{o}
• u_{a}) u_{a}
= [(r
× F) • u_{a}]
u_{a}
Here u_{a} is the unit vector in either direction of the
line a.



