and B , denoted as A . B (read . Introduction . Notions of work and kinetic energy : The work-energy theorem .
Work . Kinetic energy . Work done by a variable force . The work-energy theorem for a variable force .
The concept of potential energy . The conservation of mechanical energy . The potential energy of a spring . Power .
Collisions Summary Points to ponder Exercises A dot B ) is defined as A . B = A B cos θ ( .1a) where θ is the angle between the two vectors as shown in Fig. . (a).
Since A, B and cos θ are scalars, the dot product of A and B is a scalar quantity. Each vector, A and B , has a direction but their scalar product does not have a direction. From Eq. ( .1a), we have A .
B = A ( B cos θ ) = B ( A cos θ ) Geometrically, B cos θ is the projection of B onto A in Fig. . (b) and A cos θ is the projection of A onto B in Fig. .
(c). So, A . B is the product of the magnitude of A and the component of B along A . Alternatively, it is the product of the magnitude of B and the component of A along B .
Equation ( .1a) shows that the scalar product follows the commutative law : A . B = B . A Scalar product obeys the distributive law : A . ( B + C ) = A .
B + A . C Further, A . ( λ B ) = λ ( A . B ) where λ is a real number.
The proofs of the above equations are left to you as an exercise. For unit vectors ɵ ɵ ɵ i, j,k we have ɵ ɵ ɵ ɵ i i j j k k = ɵ ɵ ɵ ɵ i j j k k i = Given two vectors j j their scalar product is