Chapter 2. Vectors. 2.1 Vectors Scalars and Vectors


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1 Chpter 2 Vectors 2.1 Vectors Sclrs nd Vectors A vector is quntity hving both mgnitude nd direction. Emples of vector quntities re velocity, force nd position. One cn represent vector in ndimensionl spce with n rrow whose initil point is t the origin, (Figure 2.1). The mgnitude is the length of the vector. Typogrphiclly, vribles representing vectors re often written in cpitl letters, bold fce or with vector overline, A,,. The mgnitude of vector is denoted. A sclr hs only mgnitude. Emples of sclr quntities re mss, time nd speed. Vector Algebr. Two vectors re equl if they hve the sme mgnitude nd direction. The negtive of vector, denoted, is vector of the sme mgnitude s but in the opposite direction. We dd two vectors nd b by plcing the til of b t the hed of nd defining + b to be the vector with til t the origin nd hed t the hed of b. (See Figure 2.2.) The difference, b, is defined s the sum of nd the negtive of b, + ( b). The result of multiplying by sclr α is vector of mgnitude α with the sme/opposite direction if α is positive/negtive. (See Figure 2.2.) 22
2 z y Figure 2.1: Grphicl representtion of vector in three dimensions. b 2 +b  Figure 2.2: Vector rithmetic. Here re the properties of dding vectors nd multiplying them by sclr. They re evident from geometric 23
3 considertions. + b = b + α = α commuttive lws ( + b) + c = + (b + c) α(β) = (αβ) ssocitive lws α( + b) = α + αb (α + β) = α + β distributive lws Zero nd Unit Vectors. The dditive identity element for vectors is the zero vector or null vector. This is vector of mgnitude zero which is denoted s 0. A unit vector is vector of mgnitude one. If is nonzero then / is unit vector in the direction of. Unit vectors re often denoted with cret overline, ˆn. Rectngulr Unit Vectors. In n dimensionl Crtesin spce, R n, the unit vectors in the directions of the coordintes es re e 1,...e n. These re clled the rectngulr unit vectors. To cut down on subscripts, the unit vectors in three dimensionl spce re often denoted with i, j nd k. (Figure 2.3). z k j y i Figure 2.3: Rectngulr unit vectors. 24
4 Components of Vector. Consider vector with til t the origin nd hed hving the Crtesin coordintes ( 1,..., n ). We cn represent this vector s the sum of n rectngulr component vectors, = 1 e n e n. (See Figure 2.4.) Another nottion for the vector is 1,..., n. By the Pythgoren theorem, the mgnitude of the vector is = n. z i 1 2 j k 3 y Figure 2.4: Components of vector The Kronecker Delt nd Einstein Summtion Convention The Kronecker Delt tensor is defined δ ij = This nottion will be useful in our work with vectors. { 1 if i = j, 0 if i j. Consider writing vector in terms of its rectngulr components. Insted of using ellipses: = 1 e n e n, we could write the epression s sum: = n i=1 ie i. We cn shorten this nottion by leving out the sum: = i e i, where it is understood tht whenever n inde is repeted in term we sum over tht inde from 1 to n. This is the 25
5 Einstein summtion convention. A repeted inde is clled summtion inde or dummy inde. Other indices cn tke ny vlue from 1 to n nd re clled free indices. Emple Consider the mtri eqution: A = b. We cn write out the mtri nd vectors eplicitly. 11 1n 1 b =. n1 nn This tkes much less spce when we use the summtion convention. n b n Here j is summtion inde nd i is free inde. ij j = b i The Dot nd Cross Product Dot Product. The dot product or sclr product of two vectors is defined, b b cosθ, where θ is the ngle from to b. From this definition one cn derive the following properties: b = b, commuttive. α( b) = (α) b = (αb), ssocitivity of sclr multipliction. (b + c) = b + c, distributive. (See Eercise 2.1.) e i e j = δ ij. In three dimensions, this is i i = j j = k k = 1, i j = j k = k i = 0. b = i b i 1 b n b n, dot product in terms of rectngulr components. If b = 0 then either nd b re orthogonl, (perpendiculr), or one of nd b re zero. 26
6 The Angle Between Two Vectors. We cn use the dot product to find the ngle between two vectors, nd b. From the definition of the dot product, b = b cosθ. If the vectors re nonzero, then ( ) b θ = rccos. b Emple Wht is the ngle between i nd i + j? ( ) i (i + j) θ = rccos i i + j ( ) 1 = rccos 2 = π 4. Prmetric Eqution of Line. Consider line in R n tht psses through the point nd is prllel to the vector t, (tngent). A prmetric eqution of the line is = + ut, u R. Implicit Eqution of Line In 2D. Consider line in R 2 tht psses through the point nd is norml, (orthogonl, perpendiculr), to the vector n. All the lines tht re norml to n hve the property tht n is constnt, where is ny point on the line. (See Figure 2.5.) n = 0 is the line tht is norml to n nd psses through the origin. The line tht is norml to n nd psses through the point is n = n. The norml to line determines n orienttion of the line. The norml points in the direction tht is bove the line. A point b is (bove/on/below) the line if (b ) n is (positive/zero/negtive). The signed distnce of point 27
7 n=1 n=0 n= n n n=1 Figure 2.5: Eqution for line. b from the line n = n is (b ) n n. Implicit Eqution of Hyperplne. A hyperplne in R n is n n 1 dimensionl sheet which psses through given point nd is norml to given direction. In R 3 we cll this plne. Consider hyperplne tht psses through the point nd is norml to the vector n. All the hyperplnes tht re norml to n hve the property tht n is constnt, where is ny point in the hyperplne. n = 0 is the hyperplne tht is norml to n nd psses through the origin. The hyperplne tht is norml to n nd psses through the point is n = n. The norml determines n orienttion of the hyperplne. The norml points in the direction tht is bove the hyperplne. A point b is (bove/on/below) the hyperplne if (b ) n is (positive/zero/negtive). The signed 28
8 distnce of point b from the hyperplne n = n is (b ) Right nd LeftHnded Coordinte Systems. Consider rectngulr coordinte system in two dimensions. Angles re mesured from the positive is in the direction of the positive y is. There re two wys of lbeling the es. (See Figure 2.6.) In one the ngle increses in the counterclockwise direction nd in the other the ngle increses in the clockwise direction. The former is the fmilir Crtesin coordinte system. n n. y θ θ y Figure 2.6: There re two wys of lbeling the es in two dimensions. There re lso two wys of lbeling the es in threedimensionl rectngulr coordinte system. These re clled righthnded nd lefthnded coordinte systems. See Figure 2.7. Any other lbelling of the es could be rotted into one of these configurtions. The righthnded system is the one tht is used by defult. If you put your right thumb in the direction of the z is in righthnded coordinte system, then your fingers curl in the direction from the is to the y is. Cross Product. The cross product or vector product is defined, b = b sin θ n, where θ is the ngle from to b nd n is unit vector tht is orthogonl to nd b nd in the direction such tht the ordered triple of vectors, b nd n form righthnded system. 29
9 z z k k j y i i y j Figure 2.7: Right nd left hnded coordinte systems. You cn visulize the direction of b by pplying the right hnd rule. Curl the fingers of your right hnd in the direction from to b. Your thumb points in the direction of b. Wrning: Unless you re lefty, get in the hbit of putting down your pencil before pplying the right hnd rule. The dot nd cross products behve little differently. First note tht unlike the dot product, the cross product is not commuttive. The mgnitudes of b nd b re the sme, but their directions re opposite. (See Figure 2.8.) Let b = b sin θ n nd b = b sin φ m. The ngle from to b is the sme s the ngle from b to. Since {,b,n} nd {b,,m} re righthnded systems, m points in the opposite direction s n. Since b = b we sy tht the cross product is nticommuttive. Net we note tht since b = b sin θ, the mgnitude of b is the re of the prllelogrm defined by the two vectors. (See Figure 2.9.) The re of the tringle defined by two vectors is then 1 b. 2 From the definition of the cross product, one cn derive the following properties: 30
10 b b b Figure 2.8: The cross product is nticommuttive. b bsinθ b Figure 2.9: The prllelogrm nd the tringle defined by two vectors. b = b, nticommuttive. α( b) = (α) b = (αb), ssocitivity of sclr multipliction. (b + c) = b + c, distributive. ( b) c (b c). The cross product is not ssocitive. i i = j j = k k = 0. 31
11 i j = k, j k = i, k i = j. b = ( 2 b 3 3 b 2 )i + ( 3 b 1 1 b 3 )j + ( 1 b 2 2 b 1 )k = cross product in terms of rectngulr components. i j k b 1 b 2 b 3, If b = 0 then either nd b re prllel or one of or b is zero. Sclr Triple Product. Consider the volume of the prllelopiped defined by three vectors. (See Figure 2.10.) The re of the bse is b c sin θ, where θ is the ngle between b nd c. The height is cosφ, where φ is the ngle between b c nd. Thus the volume of the prllelopiped is b c sin θ cosφ. b c φ θ b c Figure 2.10: The prllelopiped defined by three vectors. Note tht (b c) = ( b c sin θ n) = b c sin θ cosφ. 32
12 Thus (b c) is the volume of the prllelopiped. (b c) is the volume or the negtive of the volume depending on whether {,b,c} is right or lefthnded system. Note tht prentheses re unnecessry in b c. There is only one wy to interpret the epression. If you did the dot product first then you would be left with the cross product of sclr nd vector which is meningless. b c is clled the sclr triple product. Plne Defined by Three Points. Three points which re not colliner define plne. Consider plne tht psses through the three points, b nd c. One wy of epressing tht the point lies in the plne is tht the vectors, b nd c re coplnr. (See Figure 2.11.) If the vectors re coplnr, then the prllelopiped defined by these three vectors will hve zero volume. We cn epress this in n eqution using the sclr triple product, ( ) (b ) (c ) = 0. b c Figure 2.11: Three points define plne. 2.2 Sets of Vectors in n Dimensions Orthogonlity. Consider two ndimensionl vectors = ( 1, 2,..., n ), y = (y 1, y 2,...,y n ). 33
13 The inner product of these vectors cn be defined y y = n i y i. i=1 The vectors re orthogonl if y = 0. The norm of vector is the length of the vector generlized to n dimensions. = Consider set of vectors { 1, 2,..., m }. If ech pir of vectors in the set is orthogonl, then the set is orthogonl. i j = 0 if i j If in ddition ech vector in the set hs norm 1, then the set is orthonorml. { 1 if i = j i j = δ ij = 0 if i j Here δ ij is known s the Kronecker delt function. Completeness. A set of n, ndimensionl vectors { 1, 2,..., n } is complete if ny ndimensionl vector cn be written s liner combintion of the vectors in the set. Tht is, ny vector y cn be written n y = c i i. i=1 34
14 Tking the inner product of ech side of this eqution with m, ( n ) y m = c i i m = i=1 n c i i m i=1 = c m m m c m = y m m 2 Thus y hs the epnsion If in ddition the set is orthonorml, then y = y = n i=1 y i i 2 i. n (y i ) i. i=1 35
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