. . Application of dot and cross product in Physics Definition . If d is the displacement vector of a particle moved from a point to another point after applying a constant force F on the particle, then the work done by the force on the particle is w F d Fig.
. If the force has an acute angle, perpendicular angle, and an obtuse angle, the work done by the force is positive, zero, and negative respectively. Example . A particle acted upon by constant forces −− is displaced from the point ( , , ) to the point ( , , ) .
Find the total work done by the forces. Resultant of the given forces is ( ) F + −− = + Let A and B be the points ( , , ) and ( , , ) respectively. Then the displacement vector of the particle is ( ) ( ) AB OB OA Therefore the work done ) ( w F d units. Fig.
. F E B D C ˆ F ˆ F ˆ F Vector - - Example . A particle is acted upon by the forces i is displaced from the point ( , , ) to the point ( , , ) . If the work done by the forces is units, find the value of λ .
Resultant of the given forces is ( ) ( F The displacement of the particle is given by ( ( ) ) As the work done by the forces is units, we have F d That is, ( ) ( ) ⇒ So, λ = − . Definition . If a force F is applied on a particle at a point with position vector r , then the torque or moment on the particle is given by F . The torque is also called the rotational force.
Fig. . Example . Find the magnitude and the direction cosines of the torque about the point ( , , ) of a force whose line of action passes through the origin.
Let A be the point ( , , ) . Then the position vector of A is OA and therefore AO Then the given force is F . So, the torque is F = −− The magnitude of the torque | = −− and the direction cosines of the torque are , , Fig. .
O F A ( , ,– ) Merry-go-round t r F F Vector - - Applications of Vector Algebra