Chapter 5 Dot, Inner and Cross Products. 5.1 Length of a vector 5.2 Dot Product 5.3 Inner Product 5.4 Cross Product
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1 Chapter 5 Dot Inner and Cross Prodcts 5. Length of a ector 5. Dot Prodct 5.3 Inner Prodct 5.4 Cross Prodct
2 5. Length and Dot Prodct in R n Length: The length of a ector ( n in R n is gien by n Notes: The length of a ector is also called its norm. Notes: Properties of length is called a nit ector. 0 iff 0 c c /40
3 Ex : (a In R 5 the length of ( 0 4 is gien by 0 ( 4 ( 5 5 (b In R 3 the length of ( is gien by ( is a nit ector 3/40
4 A standard nit ector in R n : e e e n Ex: the standard nit ector in R : the standard nit ector in R 3 : i j 0 0 i j k Notes: (Two nonzero ectors are parallel c c c 0 0 and hae the same direction and hae the opposite direction 4/40
5 Thm 5.: (Length of a scalar mltiple Let be a ector in R n and c be a scalar. Then Pf: c ( c ( c c c ( n c c c ( cn ( c c c n ( c n c ( n c c n 5/40
6 Thm 5.: (Unit ector in the direction of If is a nonzero ector in R n then the ector has length and has the same direction as. This ector is called the nit ector in the direction of. Pf: ( has the same direction as is nonzero 0 0 ( has length 6/40
7 Notes: ( The ector is called the nit ector in the direction of. ( The process of finding the nit ector in the direction of is called normalizing the ector. 7/40
8 Ex : (Finding a nit ector Find the nit ector in the direction of ( 3 and erify that this ector has length. Sol: ( (3 3 ( 4 ( is a nit ector /40
9 Distance between two ectors: The distance between two ectors and in R n is d( Notes: (Properties of distance ( ( d( 0 d( 0 if and only if (3 d( d( 9/40
10 Ex 3: (Finding the distance between two ectors The distance between =(0 and =( 0 is d( (0 0 ( 3 0/40
11 Keywords in Section 5.: طول length: معيار norm: متجه الوحدة ector: nit متجه الوحدة األساسي : ector standard nit معايرة normalizing: المسافة distance: زاوية angle: متباينة المثلث ineqality: triangle نظرية فيثاغورس theorem: Pythagorean /40
12 5. Dot Prodct Dot prodct in R n : The dot prodct of ( n and ( n is the scalar qantity n n Ex 4: (Finding the dot prodct of two ectors The dot prodct of =( 0-3 and =(3-4 is ( (3 (( (0(4 ( 3( 7 /40
13 Thm 5.3: (Properties of the dot prodct If and w are ectors in R n and c is a scalar then the following properties are tre. ( ( ( w w (3 c( ( c ( c (4 (5 0 and 0 if and only if 0 3/40
14 Eclidean n-space: R n was defined to be the set of all order n-tples of real nmbers. When R n is combined with the standard operations of ector addition scalar mltiplication ector length and the dot prodct the reslting ector space is called Eclidean n-space. 4/40
15 Ex 5: (Finding dot prodcts ( (5 8 w ( 4 3 (a (b ( w (c ( (d (e Sol: ( a ((5 ( (8 6 ( b ( w 6w 6( 4 3 (4 8 w ( w ( c ( ( ( 6 (d w w w ( 4( 4 (3(3 5 ( e w (5 ( 8 86 (3 ( w ((3 ( ( 6 4 5/40
16 Ex 6: (Using the properties of the dot prodct Sol: Gien Find ( (3 ( (3 (3 (3 (3 ( (3 ( 3( 6( ( 3( 7( ( 3(39 7( 3 ( /40
17 Thm 5.4: (The Cachy - Schwarz ineqality If and are ectors in R n then ( denotes the absolte ale of Ex 7: (An example of the Cachy - Schwarz ineqality Verify the Cachy - Schwarz ineqality for =( - 3 and =( 0 - Sol: /40
18 The angle between two ectors in R n : cos 0 Opposite direction Same direction Note: cos The angle between the zero ector and another ector is not defined. cos 0 cos 0 0 cos 0 0 cos 8/40
19 Ex 8: (Finding the angle between two ectors ( 4 0 ( 0 Sol: ( 4( (0(0 (( ( ( cos and hae opposite directions. ( 9/40
20 Orthogonal ectors: Two ectors and in R n are orthogonal if 0 Note: The ector 0 is said to be orthogonal to eery ector. 0/40
21 Ex 0: (Finding orthogonal ectors Determine all ectors in R n that are orthogonal to =(4. Sol: (4 Let (4 ( 4 0 t t ( t t t R /40
22 Thm 5.5: (The triangle ineqality Pf: If and are ectors in R n then ( ( ( ( ( ( ( Note: Eqality occrs in the triangle ineqality if and only if the ectors and hae the same direction. /40
23 Thm 5.6: (The Pythagorean theorem If and are ectors in R n then and are orthogonal if and only if 3/40
24 Dot prodct and matrix mltiplication: n n ] [ ] [ n n n n T (A ector in R n is represented as an n colmn matrix ( n 4/40
25 Keywords in Section 5.: الضرب النقطي prodct: dot فضاء نوني اقليدي n-space: Eclidean متباينة كوشي-شوارز ineqality: Cachy-Schwarz متباينة المثلث ineqality: triangle نظرية فيثاغورس theorem: Pythagorean 5/40
26 5.3 Inner Prodct Inner prodct: Let and w be ectors in a ector space V and let c be any scalar. An inner prodct on V is a fnction that associates a real nmber < > with each pair of ectors and and satisfies the following axioms. ( ( (3 w w c c (4 0 and 0 if and only if 0 6/40
27 Note: dot prodct (Eclidean inner prodct for R n general inner prodct for ector spacev Note: A ector space V with an inner prodct is called an inner prodct space. Vector space: V Inner prodct space: V 7/40
28 Ex : (The Eclidean inner prodct for R n Show that the dot prodct in R n satisfies the for axioms of an inner prodct. Sol: ( ( n n n n By Theorem 5.3 this dot prodct satisfies the reqired for axioms. Ths it is an inner prodct on R n. 8/40
29 Ex : (A different inner prodct for R n Show that the fnction defines an inner prodct on R where and. ( ( Sol: ( a w w ( ( ( ( w w w w w w ( ( w w b w 9/40
30 Note: (An inner prodct on R n 0 i n n n c c c c ( ( ( ( c c c c c c 0 ( d 0 ( /40
31 Ex 3: (A fnction that is not an inner prodct Show that the following fnction is not an inner prodct on R Sol: Let ( Then (( (( (( 6 0 Axiom 4 is not satisfied. Ths this fnction is not an inner prodct on R 3. 3/40
32 Thm 5.7: (Properties of inner prodcts Let and w be ectors in an inner prodct space V and let c be any real nmber. ( ( w w w (3 c c Norm (length of : Note: 3/40
33 Distance between and : d( Angle between two nonzero ectors and : cos 0 Orthogonal: ( and are orthogonal if 0. 33/40
34 Notes: ( If then is called a nit ector. ( 0 not a nit ector Normalizing (the nit ector in the direction of 34/40
35 Ex 6: (Finding inner prodct Sol: p q a b a b a b is an inner prodct 0 0 n n Let p( x x q( x 4 x x be polynomials in P ( x ( a p q? ( b q? ( c d( p q? ( a p q ((4 (0( ( ( ( b q q q 4 ( ( c p q 3 x 3x d( p q p q p q p q ( 3 ( 3 35/40
36 Properties of norm: ( ( 0 if and only if (3 0 0 c c Properties of distance: ( d( 0 ( d( 0 if and only if (3 d( d( 36/40
37 Thm 5.8: Let and be ectors in an inner prodct space V. ( Cachy-Schwarz ineqality: Theorem 5.4 ( Triangle ineqality: Theorem 5.5 (3 Pythagorean theorem : and are orthogonal if and only if Theorem /40
38 Orthogonal projections in inner prodct spaces: Note: Let and be two ectors in an inner prodct space V sch that 0. Then the orthogonal projection of onto is gien by proj If is a init ector then. The formla for the orthogonal projection of onto takes the following simpler form. proj 38/40
39 Ex 0: (Finding an orthogonal projection in R 3 Use the Eclidean inner prodct in R 3 to find the orthogonal projection of =(6 4 onto =( 0. Sol: (6( (( (4( proj 0 5 ( 0 ( 4 0 Note: proj (6 4 (4 0 (4 4is orthogonal to (0. 39/40
40 Thm 5.9: (Orthogonal projection and distance Let and be two ectors in an inner prodct space V sch that 0. Then d( proj d( c c 40/40
41 Keywords in Section 5.: ضرب داخلي prodct: inner فضاء الضرب الداخلي space: inner prodct معيار norm: مسافة distance: زاوية angle: متعامد orthogonal: متجه وحدة ector: nit معايرة normalizing: متباينة كوشي - شوارز ineqality: Cachy Schwarz متباينة المثلث ineqality: triangle نظرية فيثاغورس theorem: Pythagorean اسقاط عمودي projection: orthogonal 4/40
42 5.4 Cross Prodct Cross prodct in R 3 : The cross prodct of is the ector qantity w ( and ( 3 i j k w Ex : (Finding the cross prodct of two ectors The cross prodct of =( 0 and =(3-4 is w /40
43 Keywords in Section 5.4: ضرب خارجي prodct: Cross فضاء الضرب الداخلي space: inner prodct معيار norm: مسافة distance: زاوية angle: متعامد orthogonal: 43/40
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