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1、Homework #1 注意：以下題目向量是以粗體字表示，但是書寫作業時，一般向量請寫成 Av，基底向量請寫成xa 、a 等。( 例如題 P.2-1 中應寫成32zyxaaaA+=v) 請用 A4 紙書寫並裝訂妥。 Deadline: 10/22 上課繳交 P. 2-1 Given three vectors A, B, and C as follows, A = ax + ay2 az3, B = ay4 + az, C = ax5 az2, find a) aA b) |A B| c) BA d) AB e) the component of A in the direction of C
2、 f) CA(or BA) g) )(CBA and CBA)( h) CBA)( and )(CBA Solution: a) ).32(141)3(2132 222zyxzyx AaaaaaaAAa+= +=vb) .53)4(6146222=+=+=zyxaaaBAvvc) .11)3()4(20=+=BAvvd) .5 .135171411coscoso11= = = ABBA ABvv e) .2911)25(291=zxCaaACCAaAvvvvf) .10134zyxaaaCA=vv(or . 410zyxaaaBA=vv) g) .42)()(=CBACBAvvvvvvh) .
3、 5402)()()(zyxaaaBCACABCBA+=vvvvvvvvv.114455)()()(zyxaaaBACCABCBA=vvvvvvvvvP. 2-3 Two vector fields represented by A = ax Ax + ay Ay + az Az and B = axBx + ayBy + azBz, where all components may be functions of space coordinates. If these two fields are parallel to each other everywhere, what must be
4、 the relations between their components? Solution: For BAvv/ everywhere, . 0zzyyxxzyxzyxzyxBA BABABBBAAAaaaBA=vvP. 2-6 The three corners of a triangle are at P1(0, 1, 2), P2(4, 1, 3), and P3(6, 2, 5). a) Determine whether P1P2P3 is a right triangle. b) Find the area of the triangle. Solution: a) Pos
5、ition vectors of the 3 corner: 526 , 34 , 2321zyxzyxzyaaaOPaaaOPaaOP+=+= 76 , 82 ,4133221zyxzyxzxaaaPPaaaPPaaPP=+= 3213221. 0PPPPPPP= is a right triangle. b) Area of triangle1 .1721 3221=PPPP P. 2-9 Unit vectors aA and aB denote the directions of two-dimensional vectors A and B that make angles and
6、, respectively, with a reference x-axis, as shown in Fig. 2-34. a) Obtain a formula for the expansion of the consine of the difference of two angles, cos( ), by taking the scalar product BAaa . b) Obtain a formula for sin( ). Solution: ,sincosyxAaaa+= .sincosyxBaaa+= a) .sinsincoscos)cos(+=BAaa b) )
7、.sin( )sincoscos(sin0sincos0sincos=zzzyxABaaaaaaa.sincoscossin)sin(= P. 2-13 Prove by vector relations that two lines in the xy-plane (L1: ;21cybxb=+ L2: =+ybxb21c) are perpendicular if their slopes are the negative reciprocals of each other. Solution: Consider line L1: ,21cybxb=+ which has a slope
8、equal to b1/b2. Denote the shifted line passing through the origin and parallel to L1 as 0 1L: . 021=+ybxb The position vector of a point (x, y) on 0 1L is: .1yaxaryx+=vIf we introduce the vector ,21babanyx+=vwe can write the equation of 0 1L as . 01=rnvvThus the vector nvis perpendicular to ,1rvand
9、 is normal to both L1 and 0 1L. It follows that the two lines L1 and L2 are perpendicular to each other if and only if their normal vectors nvand 21babanyx+=vare orthogonal: , 0=nnvvwhich implies , 02211=+bbbb or ;2112 bb bb=that is, the slopes of lines L2 and L1 are the negative reciprocals of each
10、 other. P. 2-17 A field is expressed in spherical coordinates by E = aR(25/R2). a) find |E| and Ex at the point P( 3, 4, 5). b) Find the angle that E makes with the vector B = ax2 ay2 + az at point P. Solution: a) 21 21)5(4)3(25222=+=RRaaEv212. 0 )5(4)3(3 21222=+=xE b) (),543501zyxRaaaa+= (),2231zyx
11、BaaaBBa+=vo1115450319cos)(cos= = BRaa. P. 2-21 Given a vector function E = axy + ayx, evaluate the scalar line integral lvdE from P1(2, 1, 1) to P2(8, 2, 1) a) along the parabola x = 2y2, b) along the straight line joining the two points. Is this E a conservative field? (That is, the line integral i
12、s independent of path.) Solution: . )(2121+=PPPPxdyydxdElvva) x = 2y2, dx = 4ydy; .14)24(212221=+=dyydyydEPPlvvb) x = 6y 4, dx = 6dy; .14)46(62121=+=dyyydEPPlvv兩個沿特定路徑的線積分相同並不必然等於此場為保守場(conservative field)；但是在此問題的場量Ev是一個保守場，因為Ev可表示為xy + C的梯度，而由Null identity: 0)(V，所以Ev是一個保守場(0=Ev)。 P. 2-23 Given a sc
13、alar function ,3sin2sinzeyxV=determine a) the magnitude and the direction of the maximum rate of increase of V at the point P(1, 2, 3), b) the rate of increase V at P in the direction of the origin. Solution: a) ,3sin2sin3cos32sin3sin2cos2z zyxeyxayxayxaV +=()().043. 0026. 02363 zyzyPaaeaaV+= +=b) ; 32zyxaaaPO= ().32141zyxPOaaaa+= ().0485. 0233 31413= +=eaVPOP
