438 PRACTICAL MATHEMATICS 



If, for example, a body is made to move with uniform velocity 

 v feet per second, in a given direction, under the action of a force 

 F lb., the line of action of this force making an angle a with the 

 direction of motion, 



The displacement of the body per second = v ft. 



The work done per second is the scalar product of the force and 

 the displacement, taken in the direction of the displacement. 



Hence the work done per sec. = F v cos a ft. lb. 



212. The Mathematical Representation of a Vector. A vector 

 can be considered to be the vector sum of its horizontal and verti- 

 cal components, and the parallelogram of vectors used to find 

 this sum becomes a rectangle. 



It has already been shown in paragraph 31 that a complex 

 quantity can be represented graphically by a magnitude measured 

 in the direction of real quantities, that is, the horizontal direction, 

 and a magnitude measured in the direction of imaginary quantities, 

 that is, the vertical direction. The two directions, then, in which 

 real and imaginary quantities are measured correspond to the two 

 directions in which the horizontal and vertical components of a 

 vector are taken. A vector can therefore be represented mathe- 

 matically by a trigonometrical complex quantity, in which the 

 real part represents the horizontal component of the vector, while 

 the imaginary part represents the vertical component. 



Thus the vector = p (cos + i sin 0) 



where p is the magnitude and the direction. 



It should be noticed that if this is recognised as a standard ex- 

 pression for a vector, then all vectors should ultimately reduce to 

 this form, and this enables us to test whether this is a suitable ex- 

 pression for a vector. A velocity is a vector for which both magni- 

 tude and direction can be functions of the time ; an acceleration 

 is also a vector for which magnitude and direction can be functions 

 of the time, but an acceleration is the direct result of differentiating 

 a velocity with respect to the time. 



Thus let v be a velocity, whose magnitude p and direction are 

 both functions of t, the time. 



Then v = p (cos + i sin 0) 



dv dp . . . A \ dQ . . n 



-r- = -r (cos + i sin 0) + p ~j- ( sin + i cos 0) 



dt dt at 



/dp dQ . \ ./dp . dQ ft \ 



= ( - cos - p -j- sin + * -jT sin + /o -j- cos 

 \dt dt / \dt dt ) 



= (A cos - B sin 0) -f i (A sin + B cos 0) 



where A = ~ and B = p -j- 



dt dt 



