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,

     AB = 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 right-hand 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 Mo, then it can be shown that

     Mo = 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

     Mo = (ry Fz - rz Fy)i + (rz Fx - rx Fz)j
                        + (rx Fy - ry Fx)k

From this equation, the Cartesian components of the moment vector Mo 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.  

     Ma = (Moua) ua
          = [(r × F) • ua] ua

Here ua is the unit vector in either direction of the line a.

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