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 right-hand rule.

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.

Cross Product Direction and Sign

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.

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.