Lesson Introduction to dot-product

Introduction to dot-product

Definition Let  u  and  v  be two vectors in a plane. Then the  dot-product  (or the  scalar product)  of the vectors  u  and  v  is the real number  |u|*|v|*,  where  |u|  and  |v|  are the magnitudes  (the lengths)  of the vectors  u  and  v,  and    is the angle between the vectors. The dot-product of vectors  |u|  and  |v|  is designated as  (u,v)  often:   (u,v) = |u|*|v|*.

Figure 1.  To the definition of the dot-product  (u,v)  of vectors  u  and  v

Two vectors  u  and  v  in a plane are perpendicular if and only if their dot-product is zero: (u,v) = 0.                                                                        (1)

For any vector  u  dot-product of the vector by itself is equal to the square of the length  (of the magnitude)  of this vector:  (u,u) = .       (2)

1.  Imagine the body on the horizontal ground surface or on the horizontal floor,  which                         moves in the horizontal direction  (Figure 2). Let some force  F  be applied to the body as shown in the  Figure 2. Let the force be directed in some angle    to the horizontal plane.  Then the body is subject to the given force,  to the weight directed vertically down,  to the reaction force from the floor to the body directed vertically up,  and to the friction force acting to the body along the ground surface or along the floor and directed oppositely to the body move. Let all these forces be in equilibrium providing the body uniform movement along the surface.

Figure 2.  The body on the horizontal floor

2.  Consider a body which can move  (or is forced to move)  along a straight line                                         as shown in the  Figure 3.  You may think about a barge or a ship in the Panama channel which is pulled by a track moving aside the channel. Let the force  F  be applied to the body and compels it  (possibly in combination with other forces or constrains)  to move along this line. The work done by this force when the body makes a displacement  s  along the straight line is the scalar product of the vector  F  and the displacement vector  s: W = (F,s).             (3)

Figure 3.  The work done by a force  (a ship in the Panama channel is pulled by a track moving aside the channel,  plane view atop)

3.  If the body subjected to the acting force  F  moves along the straight or curved                                 line and has the velocity presented by the vector  u  in each time moment  (Figure 4), then the power  P  (in the physical sense)  developed by the force  F  is equal in each time moment to the scalar product of the vectors  F  and  u: P = (F,u).             (4) It does not matter whether the vectors  F  and  u  are co-directed or make some angle. The force vector  F  may vary in the magnitude and/or in the direction,  as well as the velocity vector. The formula  (4)  for the power works in all cases. If one is required to calculate the work done by the force on the curved path,  then the integral of the power  (4)  is calculated from the initial to the final time moment.

Figure 4.  The power developed by a force. The instant force vector  F  and the instant velocity vector  u  are presented.  The orbital movement