L AN INTRODUCTION TO VECTOR ARITHMETIC Introduction 2 I n t h e formula V = a r h , w e observe t h a t , t o f i n d V we must know t h e v a l u e s of both r and h. I n t e r m s of a f u n c t i o n machine, we may t h i n k of t h e i n p u t a s being t h e o r d e r e d p a i r of numbers r and h , w r i t t e n , s a y , a s ( r , h ) . The o u t p u t i s o b t a i n e d by m u l t i p l y i n g t h e square of t h e f i r s t member by t h e second member and then m u l t i p l y i n g t h e r e s u l t i n g product by n. W e s a y ordered p a i r s i n c e i t c e r t a i n l y makes a d i f f e r e n c e i n g e n e r a l whether it i s r o r h t h a t i s being squared. For example, t h e i n p u t (3,4) y i e l d s n32 4 = 36 a a s an o u t p u t , whereas ( 4 , 3 ) y i e l d s a4 2 3 = 48n a s an o u t p u t . While w e s h a l l d e a l w i t h t h i s i d e a i n g r e a t e r d e t a i l a s our course u n f o l d s , l e t us say f o r now t h a t an ordered p a i r of numbers w i l l be c a l l e d a 2-tuple and t h a t , q u i t e i n g e n e r a l , an n-tuple w i l l r e f e r t o an o r d e r e d sequence of n numbers. I n t h i s .context, t h e s t u d y of f u n c t i o n s of s e v e r a l r e a l v a r i a b l e s i n v o l v e s a s t u d y of n-tuples. \ For example, given t h e equation w e s e e t h a t t o determine y it i s r e q u i r e d what w e know t h e v a l u e s of xl, x2, x3, and x4. I n t h i s c o n t e x t , t h e y-machine, which y i e l d s y a s an o u t p u t , h a s t h e 4-tuple (xl, x2, x3, x 4 ) a s t h e i n p u t . A s a I s p e c i f i c i l l u s t r a t i o n , i f (4,2,1,6) is t h e i n p u t , t h e o u t p u t i s 42 + 2(213 + 4 ( 1 ) + 5 ( 6 ) = 66. Thus, a s we hope t o make more c l e a r a s we go along, t h e study of n - t u p l e a r i t h m e t i c i s a r a t h e r important a s p e c t of f u n c t i o n s of seve r a l variables. What i s a l s o i n t e r e s t i n g i s t h a t , i n t h e c a s e of 2-tuples and 3-tuples, t h e r e i s a r a t h e r i n t e r e s t i n g s t r u c t u r a l connection w i t h t h e u s u a l 2- and 3-dimensional v e c t o r s of p h y s i c s and engineering. While we wish t o go beyond t h e t r a d i t i o n a l 2- and 3-dimensional c a s e i s favor of t h e general n-tuple case, t h e f a c t i s t h a t ordinary vectors a r e i n t e r e s t i n g and important i n t h e i r own r i g h t , and, a t t h e same time, by g e t t i n g f a m i l i a r w i t h t h e t a n g i b l e 2- and 3-dimensional c a s e s , we b u i l d our experience s o t h a t we may b e t t e r handle t h e more a b s t r a c t n-tuple a r i t h m e t i c when n i s g r e a t e r than 3. With t h i s i n mind, w e devote t h i s c h a p t e r f i r s t t o reviewing t h e t r a d i t i o n a l i d e a of v e c t o r s and second t o showing how v e c t o r arithmet i c i s s t r u c t u r a l l y r e l a t e d t o numerical a r i t h m e t i c a s s t u d i e d i n t h e previous c h a p t e r . I n t h e n e x t c h a p t e r w e s h a l l extend o u r i d e a s t o t h a t of t h e c a l c u l u s of v e c t o r s , a f t e r which w e branch o u t t o t h e more g e n e r a l c a l c u l u s of n-dimensional space ( t h e name u s u a l l y given t o t h e a r i t h m e t i c of n-tuples) . A t any r a t e , w i t h t h e overview i n mind, we now t u r n t o a d i s c u s s i o n o f 'traditionalmvectors. Vectors R e v i s i t e d C e r t a i n q u a n t i t i e s depend on d i r e c t i o n a s w e l l a s magnitude. For example, i f t h e d i s t a n c e from town A t o town B i s 400 m i l e s and w e l e a v e A and t r a v e l a t 50 miles p e r hour w e w i l l a r r i v e a t B i n 8 hours provided we t r a v e l i n t h e d i r e c t i o n from A t o B. A q u a n t i t y which depends on d i r e c t i o n a s w e l l as magnitude i s c a l l e d a v e c t o r ( a s opposed t o a q u a n t i t y which depends o n l y bn magnitude, which i s c a l l e d a s c a l a r ) . A r a t h e r n a t u r a l q u e s t i o n is: how s h a l l w e r e p r e s e n t a v e c t o r q u a n t i t y p i c t o r i a l l y ? R e c a l l t h a t f o r s c a l a r q u a n t i t i e s we have a l r e a d y agreed t o u s e t h e number l i n e and i d e n t i f y numbers w i t h l e n g t h s . That i s , w e t h i n k of t h e number 3 a s a l e n g t h of 3 u n i t s . W e generalize t h i s idea t o represent vector quantities. Namely, we a g r e e t h a t r a t h e r than use only t h e l e n g t h of a l i n e ( a s w e do w i t h That i s , s c a l a r s ) w e w i l l u s e th'e n o t i o n of d i r e c t e d l e n g t h as w e l l . given t h e v e c t o r which we wish t o r e p r e s e n t , w e draw a l i n e i n t h e g-: r e c t i o n of t h e v e c t o r , whereupon we t h e n choose t h e l e n g t h of the l i n e t o r e p r e s e n t t h e magnitude of t h e v e c t o r . For example, i f we wish to i n d i c a t e a f o r c e of 2 pounds a c t i n g i n t h e d i r e c t i o n of t h e l i n e y = x , w e would r e p r e s e n t i t a s t h e arrow 3. - F1 L t h i s a l s o r e p r e s e n t s a f o r c e whose magnitude i s 2 pounds, b u t , w h i l e $ and P1 have e q u a l magnitudes, they a r e d i f f e r e n t f o r c e s s i n c e they a c t i n d i f f e r e n t d i r e c t i o n s . Notice from o u r diagram t h a t t h e r e i s s t i l l one i n g r e d i e n t t h a t must be d e f i n e d i f we wish t o r e p r e s e n t o u r v e c t o r w i t h o u t ambiguity. We That i s , i n our p r e s e n t example, must i n d i c a t e t h e s e n s e of t h e v e c t o r . do w e mean o r do w e mean ' both v e c t o r s have magnitude e q u a l t o 2 and both a c t i n t h e d i r e c t i o n of y = x. a We only t a l k about s e n s e once t h e d i r e c t i o n i s known. I n o t h e r words, we do n o t ask whether two v e c t o r s have t h e same s e n s e u n t i l we know whether they a c t i n t h e same d i r e c t i o n . an "arrowhead." To i n d i c a t e t h e s e n s e we use Thus, we may say t h a t t h e geometric i n t e r p r e t a t i o n o f ' a v e c t o r i s a s an arrow. I t i s important t o n o t e ( i n much t h e same v e i n a s our d i s c u s s i o n i n P a r t 1 of Calculus of t h e d i f f e r e n c e between a f u n c t i o n and i t s graph) t h a t t h e arrow i s t h e geometric rep r e s e n t a t i o n of t h e v e c t o r n S , t h e v e c t o r i t s e l f . For example, f o r c e and v e l o c i t y a r e n o t arrows b u t merely r e p r e s e n t a b l e a s arrows. Of c o u r s e , once w e have made t h i s d i s t i n c t i o n , we may use arrows and vect o r s more o r less i n t e r c h a n g e a b l y i n the'same way we use graphs and functions. I n any e v e n t , we a r e now ready t o begin our "game" of v e c t o r ( o r arrow) arithmetic. So f a r , a l l we have i s a s e t of obj.ects c a l l e d v e c t o r s . - We do n o t , a s y e t , have a s t r u c t u r e . A mathematical s t r u c t u r e i s more than a s e t . I t i s a s e t t o g e t h e r w i t h v a r i o u s r u l e s and o p e r a t i o n s . For a s t a r t , we would most l i k e l y p r e f e r t o d e f i n e some equivalence r e l a t i o n s o t h a t we can determine when two v e c t o r s can be c a l l e d e q u i v a l e n t (equal). Since t h e only important i n g r e d i a n t s of a v e c t o r a r e i t s mag- - n i t u d e , s e n s e , and d i r e c t i o n , we a g r e e t o c a l l two v e c t o r s (arrows) "equal" i f and o n l y i f they have t h e same magnitude, sense and d i r e c t i o n . In p a r t i c u l a r , t h i s d e f i n i t i o n of e q u a l i t y means t h a t i f we have two d i s t i n c t p a r a l l e l l i n e segments of t h e same l e n g t h and w e o r i e n t them w i t h t h e same s e n s e then t h e s e two arrows a r e equal. P i c t o r i a l l y , + + w r i t e A = B. Notice t h a t a s l i n e s they would be p a r a l l e l but . d i f f e r e n t . + A and B a r e c a l l e d e q u a l and w e I f t h i s usage of e q u a l i t y s t r i k e s you a s being r a t h e r odd, n o t i c e t h a t w e have used t h i s i d e a a s e a r l y , f o r example, a s when we make such 2 remarks a s 1 C l e a r l y , t h e s e two syinbols do n o t look a l i k e . What 7= we mean i s t h a t i f i s d e f i n e d t o mean t h a t number which when multip l i e d by b y i e l d s a , then 1 = 2 s i n c e t h e number which must be multi- g - - p l i e d by 2 t o y i e l d 1 i s t h e same a s t h e number which must be m u l t i p l i e d a as an exponent by 4 t o y i e l d 2 . On t h e o t h e r hand, i f we a g r e e t o use JS wherein t h e numerator t e l l s us what power t o r a i s e t o and t h e denominat o r t e l l s us t h e r o o t t o a b s t r a c t , then 1 and 2 a r e no longer equal -4 = E l ; -11, w h i l e l2I4= { I , -1, i , - i ) . I n o t h e r words, since 1 e q u a l i t y i s always w i t h r e s p e c t t o a p a r t i c u l a r l y d e f i n e d r e l a t i o n . I n any e v e n t , n o t i c e t h a t t h i s d e f i n i t i o n of t h e e q u a l i t y of two vectors i s to itself, equals t h e t h e second an equivalence r e l a t i o n . That i s , (1) any v e c t o r i s e q u a l (2) i f t h e f i r s t v e c t o r e q u a l s t h e second then t h e second f i r s t , and ( 3 ) i f t h e f i r s t v e c t o r e q u a l s t h e second and e q u a l s t h e t h i r d , then t h e f i r s t e q u a l s t h e t h i r d . Hence, i f we go back t o r u l e s E-1 through E-5 i n S e c t i o n D of Chapter 1, we s e e t h a t t h e s e r u l e s remain " t r u e " i f we everywhere r e p l a c e t h e word "numberw by t h e word "vector." (For convenience i n r e f e r r i n g , we s h a l l + + use a and b r a t h e r than a and b when r e f e r r i n g t o our 1 6 r u l e s . ) The n e x t problem i n developing our s t r u c t u r e i s t o impose a b i n a r x o p e r a t i o n on t h e s e t of v e c t o r s . This can, of course, be done i n I n o t h e r words, t h e r e a r e an e n d l e s s number of many d i f f e r e n t ways. ways i n which w e can make up r u l e s whereby we can combine v e c t o r s t o Rather than proceed randomly (and h e r e i s a n a t u r a l form v e c t o r s . connection between p u r e and a p p l i e d mathematics) w e choose a s our r u l e one t h a t w e b e l i e v e t o be t r u e i n t h e r e a l world. In particular, the p h y s i c a l n o t i o n of t h e r e s u l t a n t v e c t o r motivates us t o choose our b i n a r y o p e r a t i o n . S t a t e d i n t e r m s of arrows, we d e f i n e t h e sum of t h e - v e c t o r s a and b ( w r i t t e n a + b ) * a s follows: We s h i f t b p a r a l l e l t o i t s e l f w i t h o u t changing i t s magnitude and sense, ( n o t i c e t h a t by o u r d e f i n i t i o n of e q u a l i t y t h e "new" v e c t o r which r e s u l t s from t h i s s h i f t i s e q u a l t o b) u n t i l i t s " t a i l " c o i n c i d e s with - t h e "head" of a . Then a + b i s merely t h e arrow t h a t s t a r t s a t t h e t a i l of a and t e r m i n a t e s a t t h e head o £ b. Pictorially, - Leaving o u t t h e a and b n o t a t i o n w e can say: t o add two v e c t o r s , we p l a c e t h e t a i l of t h e second a t t h e head of t h e f i r s t , and t h e sum i s then t h e arrow which goes from t h e t a i l of t h e f i r s t t o t h e head of t h e second. Again, it i s important t o n o t e t h a t we were f o r c e d t o pick t h i s d e f i n i t i o n , b u t by making t h i s choice we a r e s u r e t h a t our d e f i n i t i o n has a t l e a s t one r e a l i n t e r p r e t a t i o n , namely t h e u s u a l i d e a of a r e s u l tant. A t t h i s s t a g e of t h e development, we a r e a t l e a s t s u r e t h a t our concept of "sum" i s a b i n a r y o p e r a t i o n s i n c e it t e l l s us how t o determine an arrow from two given arrows. W e now check t o s e e what p r o p e r t i e s our W ~ ~ has. I I Our f i r s t c o n t e n t i o n i s t h a t f o r any p a i r of v e c t o r s a and b, a + b = b + a . To check t h a t t h i s p r o p e r t y h o l d s , we need only comp a r e a + b and b + a and observe what happens. To t h i s end: a + b and b + a a r e o p p o s i t e s i d e s of a p a r a l l e l o g r a m and have t h e same sense. Therefore, a + b = b + a. i s customary t o r e f e r t o "sum" and t o use t h e symbol + t o denote a b i n a r y o p e r a t i o n . We must be broadminded enough, mathematically speaking, n o t t o t h i n k of "sum" a s being r e s t r i c t e d t o t h e o r d i n a r y s u m of two numbers. A c t u a l l y , however, i f we wished t o be completely r i g o r o u s and unambiguous, we should have invented a new symbol, f o r example, a*b r a t h e r than a + b t o denote t h e b i n a r y o p e r a t i o n whereby we combine two v e c t o r s t o form a v e c t o r . 31t . W e now check t o see whether f o r t h r e e v e c t o r s a , b , and c, a + ( b + c ) = ( a + b ) + c . We observe: t (b + c ) and (a + b) + c a + both name t h i s v e c t o r Notice, t h e r e f o r e , t h a t a + b + c i s unambiguous, and i t i s t h e v e c t o r t h a t goes from t h e t a i l of 5 t o t h e head of c when a , b , and c a r e p r o p e r l y a l i g n e d head t o t a i l . a What t h i s means i s t h a t w e do n o t have t o worry about "voice i n f l e c t i o n " when w e "add" v e c t o r s , o r , i n o t h e r words, i f a , b, c , d , and e denote v e c t o r s , t h e e x p r e s s i o n a + b + c + d + e unambiguously names a v e c t o r . P i c t o r i a l l y , t h i s becomes t h e "polygon r u l e , " a g e n e r a l i z a t i o n c f t h e parallelogram r u l e i l l u s t r a t e d e a r l i e r . Notice t h a t , s o f a r , i f w e continue t o r e p l a c e t h e word "number" by 'Vector," r u l e s A-1, A-2, and A-3 s t i l l apply t o our game of v e c t o r s . Our n e x t s t e p i s t o s e e whether t h e r e i s a v e c t o r which p l a y s a r o l e of a d d i t i v e i d e n t i t y . That i s , given any v e c t o r b , i s t h e r e a v e c t o r , I n terms of o u r arrow i n t e r p r e t a denoted by 0, such t h a t b + 0 = b. t i o n , i t i s c l e a r t h a t i f t h e v e c t o r 0 were t o have a non-zero l e n g t h then b and b + 0 could n o t p o s s i b l y have t h e same l e n g t h . Again, pictorially, t e r m i n a t e s on t h i s c i r c l e I f 0 had magnitude r # 0 t h e n b + 0 would have t o t e r m i n a t e somewhere on t h e c i r c l e . And no p o i n t on b+0 t h e c i r c l e can be t h e c e n t e r of t h e c i r c l e . + Thus, for the vector 0 to have the desired property, we must define it to be a vector whose magnitude is zero (i.e., the number O), indepen- dently of any mention of direction or sense. To be able to refer to the zero vector, (as we refer to the number zero), we must agree that we do not distinguish between two vectors of zero magnitude, even if they have different directions. This agrees with our geometric intuition, since a point has no direction. Thus, the definition of equality is waived for the zero vector, and we simply agree to call any two vectors of zero magnitude equal. Finally, we want to investigate the notion of whether, given any vector -a, we can find another vector b such that a + b = 0 (where 0 here denotes the vector O*). Since the zero vector has no length and since we add vectors "head to tail" it follows that if a + b is to equal 0 then the tail of a and the head of b must coincide. This in turn means that we have the same magnitude and direction as a but the opposite sense. Pictorially, : ;,"., : ;. . > . - - . If b originates at P it must terminate at Q if a + b = 0. Therefore, if we think of a as lib then b = m. In other words, if we wish vector addition to have the same structure as that of the numerical addition (and the choice is ours to make) we must define the inverse of 5, i.e. (-a), to be the vector which has the same magnitude and direction as a but the opposite sense. - If we again agree, as in the case of numerical addition, that a + (-b) will be abbreviated by a b, then to form the vector a - b we proceed as follows: - (1) a - b means a + (-b) (2) To obtain (-b) from b simply reverse the sense of b. ( 3 ) We now add a and (-b) in the "usual" way; the sum being a or, therefore, a - b. + (-b), Pictorially, "Because there is a possibliity of confusing scalars and vectors in many cases, it is conventional to use different symbolisms for vectors and scalars. In some texts, one uses greek letters for vectors and "regular" letters for scalars; or one uses boldface type for vectors, or one writes arrows over the vector (such as 6 ) . Later we will do this but for now we prefer to have our symbolism look as much like that in rules E-1 through A-5 as possible. I - . (ii) (iii) or a - b While (we hopei) that our explanation of subtraction is adequate and that i t certainly shows the resemblance between the relationship, structurally, of vector subtraction and numerical subtraction, there is yet another way to view subtraction of vectors a way that might be more easily remembered from a computational point of view. - Suppose we are given the vectors a and b and we now place them tail-to-tail. (If w e desire a rationalization, if we add head-to-tail, why not subtract tail-to-tail?) Let us now look at the vector that extends from the head of b to the head of a, and just as in numerical arithmetic, let us label this "unknown" x. Thus, - In the above arrangement of vectors, only x and b are properly aligned' (head-to-tail) for addition. That is, our diagram yields the "equationn; 7 Had we not been told that (1) was a vector equation, and instead we treated equation (1) w l y numbers were involved, we would have ., ".:.' obtained I n numerical a r i t h m e t i c , - t h e process of g e t t i n g from (1) t o ( 2 ) i s c a l l e d t r a n s p o s i n g , and i t s v a l i d i t y i s ' e s t a b l i s h e d roughly along t h e I t turns out t h a t l i n e s of e q u a l s s u b t r a c t e d from e q u a l s a r e equal. t h i s p r o p e r t y of t r a n s p o s i n g which g o t .us from (1) t o (2). i s a property of o u r f i v e r u l e s f o r e q u a l i t y and"our f i v e r u l e s f o r a d d i t i o n . [The proof i s l e f t t o t h e i n t e r e s t e d r e a d e r , and i n v o l v e s w r i t i n g x a s (x + b) + (-b) = a + (-b). I + b = a Since t h e s e same t e n r u l e s apply t o v e c t o r s a s w e l l a s numbers, our game-idea t e l l s us t h a t t h e process of g'etting from (1) t o ( 2 ) i s e q u a l l y v a l i d when w e a r e d e a l i n g with vectors. Summed up, then p i c t o r i a l l y , t o f i n d t h e d i f f e r e n c e of two v e c t o r s , w e p l a c e t h e two v e c t o r s t a i l - t o - t a i l , and then draw a v e c t o r from t h e head of one t o t h e head of t h e o t h e r . Notice t h a t t h i s can be done w i t h two d i f f e r e n t s e n s e s , b u t t h e r u l e of t r a n s p o s i n g t e l l s us which i s which. That is: therefore x = a - b Summed up more f o r m a l l y , i f a and b a r e a n y two v e c t o r s and we wish t o - b, we p l a c e t h e two v e c t o r s t a i l - t o - t a i l and a find a t h e v e c t o r which goes from t h e head of b , t o t h e head of - b i s then 5. Again we must stress t h e importance of s t r u c t u r e . The mere f a c t t h a t we have a b i n a r y o p e r a t i o n denoted by "+" and a s t a t e m e n t t h a t a + b = c , we must n o t jump t o t h e conclusion t h a t a = c b. Transposing i s a theorem i n a s t r u c t u r e t h a t obeys some p a r t i c u l a r r u l e s . A s an example, l e t us a g a i n t a k e "+" t o mean union. Then i f a + b = c , - it i s not necessarily true t h a t a = c C = AUB (A + B ) - b. Pictorially, therefore, C -B= Again, s i n c e t h e s t r u c t u r e of sets w i t h r e s p e c t t o union i s d i f f e r e n t from t h e s t r u c t u r e of numbers w i t h r e s p e c t t o a d d i t i o n , t h i s example i s n o t any kind of c o n t r a d i c t i o n . Rather it should c a u t i o n us t h a t while it i s n i c e t o t r a n s p o s e j u s t because it seems " n a t u r a l , " w e must n o t b e l i t t l e t h e f a c t t h a t it i s only because of a p a r t i c u l a r s t r u c t u r e t h a t w e can enjoy t h i s p r i v i l e g e . With t h e s e remarks behind us, w e a r e now i n a p o s i t i o n t o see how t h e s t r u c t u r e of a r i t h m e t i c r e a l l y works. Notice t h a t i n t h e playing of our "game" it i s t h e r u l e s which t e l l us how t h e terms a r e r e l a t e d t h a t a r e s o important n o t t h e terms themselves. For example, w e have j u s t seen t h a t r u l e s E-1 through A-5 which a p p l i e d t o n o r d i n a r y " a r i t h m e t i c a r e a l s o c o r r e c t f o r v e c t o r a r i t h m e t i c provided t h a t everywhere t h e word "number" appears w e s u b s t i t u t e t h e word "vector." Consequently, any conclusion t h a t follows inescapably from t h e s e t e n r u l e s i n numerical a r i t h m e t i c w i l l be a v a l i d conclusion i n v e c t o r arithmetic a s w e l l . - one way t o i l l u s t r a t e t h i s i d e a i s t o t a k e a proof which w e have given i n numerical a r i t h m e t i c and reproduce i t verbatim, b u t r e p l a c e "nurnbePLby " v e c t o r , " and observe t h a t t h e given proof i s s t i l l v a l i d i n t h e new s i t u a t i o n . More s p e c i f - i c a l l y , Theorem 1 I f a , b , and c a r e v e c t o r s such t h a t a + b = a + c then b = c. P roof Statement Reason (1) There e x i s t s a v e c t o r -a (1) A-5 + ( a + b) (3) -a + ( a + b) -a + ( a + C ) ( 4 ) (-a + a ) + b (5) -a + a = a + ( 6 ) a + (-a) = 0 (2) -a = -a + (a + C) = (-a = (-a + = (-a + a) + c + a) a) + + (2) b (3) c E-4 (replacing a + b by a + c) A-3 ( 4 ) S u b s t i t u t i n g (E-4) ( 3 ) i n t o (2) (5) A-2 (6) A-5 (-a) Proof cont'd Statement Reason , (7) Substituting (E-4) (5) into (6) ( 8 ) Substituting (E-4) ( 7 ) into (4) +,I + 0= c + o = o + c (9) b "1 (10) b + 0 = c + O = c (11) O + b = b O + c = c (9) A-2 (10) A-4 (11) Substituting (E-4) (9) into (10) -> / (12) Substituting (E-4) (11) into (8) (12) b = c q.e.d. Notice that we copied that statement-reason proof word for word from the corresponding theorem in numerical arithmetic. However, if we were to show this proof to a person without telling him how we obtained it, the proof stands validly on its own without reference to the proof from which we copied it. This idea can be quite readily generalized as follows. Let S denote a set and suppose "=" denotes any equivalence relation defined on S .r ,-, while "+" denotes any binary operation on S which obeys A-2 through: A-5. (We omit A-1 since A-1 is automatically obeyed by virtue of the fact that "+" denotes a binary operation. That is, we defined a binary operation to be equivalent to the Rule of Closure.) Then, in this . . particular structure, if a, b, and c are any elements in S and if ,. a + b = a + c, then b = c.* This result is an inescapable conclusion based on the assumptions encompassed by the ten rules E-1 through A-5. (In fact, if we want to be even more precise, we can argue that the result follows from a subset of these ten rules, since not all ten rules were used in the proof.) ' *We cannot emphasize enough the idea that the structure must be the same. That is, our cancellation law depends on our ten rules. AS a counter-example, suppose we think of "+" as denoting the union of sets. Then, if a, b, and c are sets, we cannot conclude that if a + b = a + c then b = c. Indeed, in our exercises in part 1 of our course we showed by example that this need not be true. (As a quick review, suppose b and c are une ual subsets of a. Then a + b = a + c since both a, but b f & i s does not contradict what we are saying equal above. Rather, structurally, the union of sets is not the same as the addition of numbers. C Scalar Multiplication Up to now we have been emphasizing the resemblance between vector and scalar arithmetic. The fact that these two different structures resem- ble one another in part is no reason to suppose that they share all properties in common, and one rather elementary but interesting differ- ence involves the process known as scalar multiplication whereby one multiplies a number (scalar) by a vector. The basic definition is pretty much straightforward. If we multiply a vector by a number c the result is a vector whose direction is that of the original vector and whose sense is the same if c is positive but opposite if c is negative. The magnitude of the new vector is times the magnitude of the original vector. By way of illustra- tion, -2v is a vector whose magnitude is twice that of v and which has the same direction but opposite sense of v. IcI Structurally, the arithmetic properties of scalar multiplication that are of the most interest to us are: SM-1 If r and s are numbers and v is a vector then r (sv) = (rs) v SM-2 If r and s are numbers ana v is a vector then (r SM-3 If r is a number and v and w are vectors then r(v SM-4 If r is any vector then lr = r. + s)v = rv + sv. + w) = rv + rw. In terms of our game of vectors (I), (2), ( 3 ) and (4) are the rules by which we play the game of scalar multiplication. That these rules are realistic follows from the fact that our model made of "arrowsp obeys them. For example (3) is the geometric equivalent of similar triangles. By way of illustration we demonstrate geometrically that 2(v + W) = 2v + 2w Other d e t a i l s a r e l e f t t o t h e textbook and t h e e x e r c i s e s . The major p o i n t t o observe i s t h a t our d i s c u s s i o n of v e c t o r s , a t l e a s t s o f a r , i s completely independent of any c o o r d i n a t e system. W e should p o i n t o u t t h a t i t i s o f t e n d e s i r a b l e t o study v e c t o r s i n C a r t e s i a n c o o r d i n a t e s s i n c e i n t h i s system t h e r e i s an i n t e r e s t i n g and r a t h e r simple form t h a t t h e a r i t h m e t i c of v e c t o r s t a k e s on. This i s explained very w e l l i n t h e t e x t and we e x p l o i t t h e s e p r o p e r t i e s i? o u r e x e r c i s e s . But it i s important t o understand t h a t t h e concept of v e c t o r s transcends any c o o r d i n a t e system. MIT OpenCourseWare http://ocw.mit.edu Resource: Calculus Revisited: Multivariable Calculus Prof. Herbert Gross The following may not correspond to a particular course on MIT OpenCourseWare, but has been provided by the author as an individual learning resource. For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.