Lecture 7: 3.2 Norm, Dot Product, and Distance in R n
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1 Lectre 7: 3. Norm, Dot Prodct, nd Distnce in R n Wei-T Ch 010/10/08
2 Annoncement Office hors: Qiz Mondy, Tesdy, nd Fridy fternoon TA: R301B, 王星翰, 蔡雅如 Sec. 1.5~1.7 8:45.m., Oct. 13, 010
3 Definitions Let nd v be two nonzero vectors in -spce or 3-spce, nd ssme these vectors hve been positioned so their initil points coincided. By the ngle between nd v, we shll men the ngle determined by nd v tht stisfies 0. If nd v re vectors in -spce or 3-spce nd is the ngle between nd v, then the dot prodct ( 點積 ) or Ecliden inner prodct ( 內積 ) v is defined by v 0 v cos if if 0 nd 0 or v 0 v 0 3
4 Dot Prodct If the vectors nd v re nonzero nd is the ngle between them, then is cte ( 銳角 ) if nd only if v > 0 is obtse ( 鈍角 )if nd only if v < 0 = / ( 直角 ) if nd only if v = 0 4
5 5 Exmple If the ngle between the vectors = (0,0,1) nd v = (0,,) is 45, then cos v v ),, ( ),, ( v v v v v v v cos v v
6 Exmple Find the ngle between digonl of cbe nd one of its edges (0,0,k) 3 d (k,k,k) 1 (k,0,0) (0,k,0) 6
7 Component Form of Dot Prodct Let =( 1,, 3 ) nd v=(v 1,v,v 3 ) be two nonzero vectors. According to the lw of cosine P Q lw of cosine 7
8 Component Form of Dot Prodct P Q 8
9 Definition If =( 1,,, n ) nd v=(v 1,v,,v n ) re vectors in R n, then the dot prodct (lso clled the Ecliden inner prodct) of nd v is denoted by.v nd is defined by Exmple: =(-1,3,5,7) nd v=(-3,-4,1,0).v = (-1)(-3) + (3)(-4) + (5)(1) + (7)(0) = -4 9
10 Theorems v 0 v cos if 0 nd v 0 if 0 or v 0 The specil cse = v, we obtin the reltionship Theorem 3.. If, v nd w re vectors in - or 3-spce, nd k is sclr, then v = v [symmetry property] (v + w) = v + w [distribtive property] k( v) = (k) v = (kv) [homogeneity property] v v 0 nd v v = 0 if v = 0 [positivity property] 10
11 Proof of Theorem 3.. k( v) = (k) v = (kv) Let =( 1,, 3 ) nd v=(v 1,v,v 3 ) 11
12 Theorem 3..3 If, v, nd w re vectors in R n, nd if k is sclr, then 0.v = v.0 = 0 (+v).w =.w + v.w.(v-w) =.v.w (-v).w =.w - v.w k(.v) =.(kv) Proof(b) [by symmetry] [by distribtivity] [by symmetry] 1
13 Exmple Clclting with dot prodcts ( - v).(3 + 4v) =.(3 + 4v) v.(3 + 4v) =3(.) + 4(.v) 6(v.) 8(v.v) =3 (.v) 8 v 13
14 Cchy-Schwrz Ineqlity With the forml The inverse cosine is not defined nless its rgment stisfies the ineqlities Fortntely, these ineqlities do hold for ll nonzero vectors in R n s reslt of Cchy-Schwrz ineqlity 14
15 Theorem.3.4 Cchy-Schwrz Ineqlity If = ( 1,,, n ) nd v=(v 1,v,,v n ) re vectors in R n, then.v v or in terms of components 15
16 To show Cchy-Schwrz Ineqlity: If = ( 1,,, n ) nd v=(v 1,v,,v n ) re vectors in R n, then.v v 16
17 Geometry in R n The sm of the lengths of two side of tringle is t lest s lrge s the third The shortest distnce between two points is stright line Theorem 3..5 If, v, nd w re vectors in R n, nd k is ny sclr, then +v + v d(,v) d(,w) + d(w,v) +v v 17
18 Proof of Theorem 3..5 Proof () Property of bsolte vle Cchy-Schwrz ineqlity Proof (b) 18
19 Theorem 3..6 Prllelogrm Eqtion for Vectors If nd v re vectors in R n, then +v + -v = ( + v ) Proof: v -v +v 19
20 Theorem 3..7 If nd v re vectors in R n with the Ecliden inner prodct, then Proof: 0
21 Dot Prodcts s Mtrix Mltipliction View nd v s colmn mtrices Exmple: 1
22 Dot Prodcts s Mtrix Mltipliction If A is n n n mtrix nd nd v re n 1 mtrices Yo cn check
23 Dot Prodct View of Mtrix Mltipliction If A=[ ij ] is m r mtrix, nd B=[b ij ] is n r n mtrix, then the ijth entry of AB is which is the dot prodct of the ith row vector of A nd the jth colmn vector of B 3
24 Dot Prodct View of Mtrix Mltipliction If the row vectors of A re r 1, r,, r m nd the colmn vectors of B re c 1, c,, c n, then the mtrix prodct AB cn be expressed s 4
25 Lectre 7: 3.3 Orthogonlity Wei-T Ch 010/10/08
26 Orthogonl Vectors Recll tht It follows tht if nd only if.v = 0 Definition: Two nonzero vectors nd v in R n re sid to be orthogonl [ 正交 ] (or perpendiclr [ 垂直 ]) if.v = 0. The zero vector in R n is orthogonl to every vector in R n. A nonempty set of vectors in R n is clled n orthogonl set if ll pirs of distinct vectors in the set re orthogonl. An orthogonl set of nit vectors is clled n orthonorml set. 6
27 Exmple Show tht =(-,3,1,4) nd v=(1,,0,-1) re orthogonl Show tht the set S={i,j,k} of stndrd nit vectors is n orthogonl set in R 3 We mst show 7
28 Norml One wy of specifying slope nd inclintion is the se nonzero vector n, clled norml ( 法向量 ) tht is orthogonl to the line or plne. The line throgh the point (x 0,y 0 ) hs norml n=(,b) y Exmple: the eqtion 6(x-3) + (y+7) = 0 represents the line throgh (3,-7) with norml n=(6,1) P(x,y) P 0 (x 0,y 0 ) n (,b) x 8
29 Theorem If nd b re constnts tht re not both zero, then n eqtion of the form x+by+c = 0 represents line in R with norml n=(,b) If, b, nd c re constnts tht re not ll zero, then n eqtion of the form x+by+cz+d = 0 represents line in R 3 with norml n=(,b,c) 9
30 Exmple The eqtion x+by=0 represents line throgh the origin in R. Show tht the vector n=(,b) is orthogonl to the line, tht is, orthogonl to every vector long the line. Soltion: Rewrite the eqtion s Therefore, the vector n is orthogonl to every vector (x,y) on the line. 30
31 An Orthogonl Projection To "decompose" vector into sm of two terms, one prllel to specified nonzero vector nd the other perpendiclr to. We hve w = w 1 nd w 1 + w = w 1 + ( w 1 ) = The vector w 1 is clled the orthogonl projection ( 正交投影 ) of on or sometimes the vector component ( 分向量 ) of long, nd denoted by proj The vector w is clled the vector component of orthogonl to, nd denoted by w = proj 31
32 Theorem 3.3. Projection Theorem If nd re vectors in R n, nd if 0, then cn be expressed in exctly one wy in the form =w 1 +w, where w 1 is sclr mltiple of nd w is orthogonl to. Proof: Since w 1 is to be sclr mltiple of, it hs the form: w 1 = k Or gol is to find vle of k nd vector w tht is orthogonl to sch tht =w 1 +w. Rewrite =w 1 +w =k+w, nd then pplying Theorems 3.. nd 3..3 to obtin.=(k+w ).=k +(w.) Since w is orthogonl to,. = k, from which we obtin Therefore, we cn get 3
33 Projection Theorem w w 1 proj proj The vector w 1 is clled the orthogonl projection of on, or the vector component of long. The vector w is clled the vector component of orthogonl to. proj proj (vector component of (vector component of long ) orthogonl to ) 33
34 Exmple L 1 Find the orthogonl projections of the vectors e 1 =(1,0) nd e =(0,1) on the line L tht mkes n ngle θwith the positive x-xis in R. Soltion: is nit vector long L. Find orthogonl projection of e 1 long. 34
35 35 Exmple Soltion: zero. is prodct dot their tht by showing perpendic lr re nd vector Verify tht the ),, ( ),, ( 1,3) (, is to orthogonl of component vector the nd ),, ( 1,) (4, is long of component vector the Ths, 1 1) ( 4 15 (3)() 1) 1)( ( ()(4) to. orthogonl of component vector the nd long of component vector the Find 1,). (4, nd 1,3) (, Let proj proj proj proj proj
36 Length of Orthogonl Projection sclr Theorem 3..1 Since If denotes the ngle between nd, then 36
37 Length of Orthogonl Projection 37
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