Elementary Linear Algebra
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1 Elementry Liner Algebr Anton & Rorres, 9 th Edition Lectre Set Chpter : Vectors in -Spce nd -Spce
2 Chpter Content Introdction to Vectors (Geometric Norm of Vector; Vector Arithmetic Dot Prodct; Projections Cross Prodct Lines nd Plnes in -Spce 9//8 Elementry Liner Algebr
3 Definitions If nd re ny to ectors, then the sm is the ector determined s follos: Position the ector so tht its initil point coincides ith the terminl point of. The ector is represented by the rro from the initil point of to the terminl point of. - If nd re ny to ectors, then the difference of from is defined by (-. - 9//8 Elementry Liner Algebr
4 Definitions If is nonzero ector nd k is nonzero rel nmber (sclr, then the prodct k is defined to be the ector hose length is k times the length of nd hose direction is the sme s tht of if k > nd opposite to tht of if k <. k if k or. A ector of the form k is clled sclr mltiple. 9//8 Elementry Liner Algebr 4
5 Remrks Rectnglr coordinte systems in dimensionl spce fll into to ctegories, left-hnded nd righthnded. 9//8 Elementry Liner Algebr 5
6 Trnsltion of Axes trnslted the xes of n xy-coordinte system to obtin n x y - coordinte system hose O is t point (x, y (k,l. A point P in -spce no hs both (x, y coordintes nd (x, y coordintes. x x k, y y l, these formls re clled the trnsltion eqtions. 9//8 Elementry Liner Algebr 6
7 Theorem.. (Properties of Vector Arithmetic If, nd re ectors in - or -spce nd k nd l re sclrs, then the folloing reltionships hold. ( ( (- k(l (kl k( k k (k l k l 9//8 Elementry Liner Algebr 7
8 Norm of Vector The length of ector is often clled the norm of nd is denoted by. 9//8 Elementry Liner Algebr 8
9 Norm of Vector the norm of ector (,, in -spce is A ector of norm is clled nit ector. The distnce beteen to points is the norm of the ector. The length of the ector k k k. 9//8 Elementry Liner Algebr 9
10 Definitions If nd re ectors in -spce or -spce nd θ is the ngle beteen nd, then the dot prodct or Ecliden inner prodct is defined by cosθ if if nd or Ө 9//8 Elementry Liner Algebr
11 9//8 Elementry Liner Algebr Exmple If the ngle beteen the ectors (,, nd (,, is 45, then 4 4 cos θ,, (,, ( 4 4 cos θ
12 Theorems Theorem.. Let nd be ectors in - or -spce. ; tht is, ( ½ If the ectors nd re nonzero nd θ is the ngle beteen them, then θ is cte if nd only if > θ is obtse if nd only if < θ π/ if nd only if 9//8 Elementry Liner Algebr
13 Theorems cosθ if nd if or Theorem.. (Properties of the Dot Prodct If, nd re ectors in - or -spce, nd k is sclr, then ( k( (k (k > if, nd if 9//8 Elementry Liner Algebr
14 Orthogonl Vectors Definition Perpendiclr ectors re lso clled orthogonl ectors. To nonzero ectors re orthogonl if nd only if their dot prodct is zero. To indicte tht nd re orthogonl ectors e rite. 9//8 Elementry Liner Algebr 4
15 An Orthogonl Projection To "decompose" ector into sm of to terms, one prllel to specified nonzero ector nd the other perpendiclr to. 9//8 Elementry Liner Algebr 5
16 Theorem.. proj proj If nd re ectors in -spce or -spce nd if, then proj proj proj cosθ (ector component of long (ector component of orthogonl to Ө 9//8 Elementry Liner Algebr 6
17 Exmple Let (,, nd (4,,. Find the ector component of long nd the ector component of orthogonl to. Ths, nd the the Verify th t the 4 ector component ector component ector their dot prodct is proj proj ((4 ( ( (( 5 ( zero. of of 5 proj long orthogonl (,, ( 7, nd 5 7 (4,,, ( is 7 7 to re, 5 7, is ( 7 6 7, 7, 7 perpendic lr by shoing tht 9//8 Elementry Liner Algebr 7
18 line L : x by c. let to points Then (x - x, y (x - y, y (x, y on L n( (, b P n (x.y Q(x.y 9//8 Elementry Liner Algebr 8
19 Qiz Find forml for the distnce D beteen point P ( x, y nd the line x by c. Let Q ( x, y D be ny proj QP QP cos θ n point on the line QP QP QP n n Bt QP QP n ( x ( x x, y x y, b( y y, P n n b Q(x.y 9//8 Elementry Liner Algebr
20 Exmple P n Soltion: (cont. Q so tht ( x x b( y y D ( b Since the point Q( x, y lies on the line, its coordintes stisfy the eqtion of the line, so x by c or c x Sbstitting this expression in ( yields the forml x by c D ( b by 9//8 Elementry Liner Algebr
21 Cross Prodct Definition If (,, nd (,, re ectors in -spce, then the cross prodct is the ector defined by 9//8 Elementry Liner Algebr
22 9//8 Elementry Liner Algebr Cross Prodct Definition If (,, nd (,, re ectors in -spce, then the cross prodct is the ector (,, defined by Tht is, Remrk,, (,,
23 Are of Prllelogrm Theorem.4. (Are of Prllelogrm If nd re ectors in -spce, then is eql to the re of the prllelogrm determined by nd. Exmple Find the re of the tringle determined by the point (,,, (-,,, nd (,4,. 9//8 Elementry Liner Algebr
24 Theorems Theorem.4. (Reltionships Inoling Cross Prodct nd Dot Prodct If, nd re ectors in -spce, then ( ( ( (Lgrnge s identity ( ( ( ( ( ( x 9//8 Elementry Liner Algebr 4
25 Theorems Theorem.4. (Properties of Cross Prodct If, nd re ny ectors in -spce nd k is ny sclr, then - ( ( ( k( (k (k 9//8 Elementry Liner Algebr
26 9//8 Elementry Liner Algebr 6 Triple Prodct ( * * *
27 9//8 Elementry Liner Algebr 7 Triple Prodct Definition If, nd re ectors in -spce, then ( is clled the sclr triple prodct of, nd. ( x
28 9//8 Elementry Liner Algebr 8 Theorem.4.4 The bsolte le of the determinnt is eql to the olme of the prllelepiped in -spce determined by the ectors (,,, (,,, nd (,,, ( * ( ( ( * * * det
29 Theorem.4.5 If the ectors (,,, (,,, nd (,, he the sme initil point, then they lie in the sme plne if nd only if ( 9//8 Elementry Liner Algebr 9
30 Plnes in -Spce One cn specify plne in -spce by giing its inclintion nd specifying one of its points. 9//8 Elementry Liner Algebr
31 Plnes in -Spce A conenient method for plne is to specify nonzero ector, clled norml, tht is perpendiclr to the plne. n (, b, c n P P The point-norml form of the eqtion of plne: (x-x b(y-y c(z-z 9//8 Elementry Liner Algebr
32 Exmple 9//8 Elementry Liner Algebr
33 Theorem.5. If, b, c, nd d re constnts nd, b, nd c re not ll zero, then the grph of the eqtion x by cz d is plne hing the ector n (, b, c s norml. (x-x b(y-y c(z-z Remrk: The boe eqtion is liner eqtion in x, y, nd z; it is clled the generl form of the eqtion of plne. 9//8 Elementry Liner Algebr
34 Theorem.5. Theorem.5. (Distnce beteen Point nd Plne The distnce D beteen point P (x,y,z nd the plne x by cz d is D x by b cz c d P n Q 9//8 Elementry Liner Algebr 4
35 9//8 Elementry Liner Algebr 5 Theorem.5. Theorem.5. (Distnce beteen Point nd Plne The distnce D beteen point P (x,y,z nd the plne x by cz d is ( ( (,, ( c b d cz by x c b z z c y y b x x c b c b P Q D Q Q Q Q P n
36 Exmple (Distnce Beteen Pont nd Plne 9//8 Elementry Liner Algebr 6
37 The Soltion of System in -Spce 9//8 Elementry Liner Algebr 7
38 Exmple 9//8 Elementry Liner Algebr 8
39 Line in -Spce Sppose tht l is the line in -spce throgh the point P (x,y,z nd prllel to the nonzero ector (, b, c. 9//8 Elementry Liner Algebr 9
40 Line in -Spce Eery point P(x, y, z on l (x, y, z P t* (x,y,z t*(, b, c 9//8 Elementry Liner Algebr 4
41 Exmple A line prllel to gien ector 9//8 Elementry Liner Algebr 4
42 Exmple (Distnce Beteen Prllel Plnes 9//8 Elementry Liner Algebr 4
43 Exmple (Distnce Beteen Prllel Plnes 9//8 Elementry Liner Algebr 4
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