390 Mr. H. T. Flint on Integration Theorems 



A vector is a quantity which may be represented by a 

 straight line in this wa} r . 



We assume four directions in space and denote them by 

 h h h h an d an Y other vector p may then be expressed in 

 terms of them : 



p = xi x + yi 2 + si 3 + ui A . 



If ii i 2 i s 2 4 are unit vectors, a y z u are the projections of p 

 along them. 



§ 2. If two vectors pi and p 2 are multiplied an expression 

 results containing terms in i m i n and i m i m , where m^=n. 



We define the product of the two like vectors i m i m to be 

 — 1, and the product of pip 2 then contains terms in two 

 vectors and other terms from which the i's have disappeared. 



We write : 



plp 2 = V2plP2 + VoPlP2 (^'l) 



V 2 /3ij0 2 contains i 2 h, i±i 2 etc. and Y pip 2 denotes the remainder 

 of the product not containing any i's. ^oP\p2 with a minus 

 sign prefixed is called the scalar product of p Y p 2 . 



If this vanishes pi and p 2 are said to be perpendicular. 

 We complete our definition of the product p^ 2 by writing 



p2Pi=— V2P1P2 + V0P1P2 (2*2) 



This gives a rule of multiplication known in the case of 

 ordinary vector analysis. It is not commutative but is 

 defined to be associative and distributive. 



In choosing the unit vectors i x i 2 i % i 4 as fundamental 

 directions we make them mutually perpendicular, 



i.e. V ^3 = etc. 



The rule of multiplication then shows that 



If we write : 



Pi = «i*i + hh + <¥3 + dih, ^ „ 



p 2 = a 2 '%i + b 2 i 2 + r 2 i s -f d 2 i±, J 

 we have on expansion : 



V /Oi/o 2 = — (a { a 2 + M2 + c x c 2 + d x d 2 ) 

 and 



Y 2 Pip 2 = i2h(^i c 2 — biCi) 4- igtj (e l a 2 — e 2 a^) + i^ 2 (aj>2 — a 2 b x ) 



+ i 1 i i (a 1 d 2 — a 2 d 1 ) +i 2 ? 4 (M 2 -y i ) + Hhi c i c h — ^i-)- 



. . (2-4) 



