MULTIPLICATION OF VECTORS 443 



Now velocity = p (cos + i sin 0) 



and acceleration - VG?) + (f ~dt}' ^ cos (^ + + * sin ( + a ) } 



dQ 



where tan a = p -j- 



3F 



-ft) = V73-17 2 + 51-33 2 

 = 89-37 



51-33 



tana= 7l7 



a = 35 3' 



Hence when t = 5-3 the magnitude of the acceleration = 89-37 

 f.s.s., and the direction = 20 36' + 35 3' = 55 39'. 



214. The Multiplication of Vectors. If A is a vector of magni- 

 tude P! and direction O x 



then & = Pi (cos X + i sin X ). 



Also if B is a vector of magnitude /> 2 and direction 2 



then B = p z (cos 2 + i sin 2 ). 



The product AB = p^ 2 (cos X + i sin X ) (cos 2 + i sin 2 ) 

 (cos (0 X + 2) + i sin (0j + 2 ) } 

 (cos (a + 20 X ) + i sin (a + 20 1 ) } 



where a = 2 - 0j, the angle between the vectors, and X is the 

 inclination of the line of action of vector A with the initial line. 



If Q l = o, that is, the initial line is so chosen that it coincides 

 with the line of action of vector A, 



then the product AB = p^ z (cos a + i sin a). 



This is an expression in the form of a complex quantity, the 

 real part being p^ z cos a and the imaginary part p^ z sin a. 

 Now the imaginary part represents a quantity which must be 

 measured in a direction perpendicular to the direction in which 

 the corresponding real part is measured. It has already been 

 shown that the real part, the product p^p z cos a, is the scalar pro- 

 duct of the two vectors taken in the direction of either of the vec- 

 tors. Hence the product p^p z sin a must be taken in a direction 

 which is perpendicular to the lines of action of the two vectors. 

 That is, this product must be taken in a direction which is per- 

 pendicular to the plane containing the lines of action of the two 

 vectors. 



