 Ch 4. Moments/Equivalent Systems Multimedia Engineering Statics Moment2-D Scalar Moment3-D Scalar Moment3-D Vector Couples and Equiv. System Distributed Loads, Intro
 Chapter 1. Basics 2. Vectors 3. Forces 4. Moments 5. Rigid Bodies 6. Structures 7. Centroids/Inertia 8. Internal Loads 9. Friction 10. Work & Energy Appendix Basic Math Units Sections Search eBooks Dynamics Statics Mechanics Fluids Thermodynamics Math Author(s): Kurt Gramoll ©Kurt Gramoll 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 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 = (Mo • ua) ua           = [(r × F) • ua] ua Here ua is the unit vector in either direction of the line a.

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