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1、Chapter 12 Vectors and Geometry of Space 12.1 Three-Dimensional Coordinate Systems*12.2 Vectors*12.3 The Dot Product*12.4 The Cross Product*12.5 Equations of Lines and Planes 12.6 Cylinders and Quadric Surfaces*12.7 Cylindrical and Spherical Coordinates,佐祈辙贵岳墒氟盖汐混誉矮赖青致苗巢村拴雍鹤吹入恼遮入育迭帐楞苍哉微积分教学资料chapter
2、12微积分教学资料chapter12,In this chapter we introduce vectors andcoordinate systems for three-dimensionalspace.This will be the setting for our study of the calculus of functions of two variablesin Chapter 14 because the graph of such afunction is a surface in space.In this chapter we will see that vector
3、s provide particularly simple descriptions of lines and planes in space.,许陕黎烦己蚂谍蚀策荐繁垣谭桶牛稳沼曹稿瘫庆淘常埃辱嗓订肄匝刹雹杉微积分教学资料chapter12微积分教学资料chapter12,12.1 Three-Dimensional Coordinate Systems Coordinate axes,Coordinate planes,xz-plane,origin O,Through point O,three axes vertical each other,by right-hand rule,we
4、 obtain a Three-Dimensional Rectangular Coordinate Systems,洋尽严题福偶与蜜踏吐豌要碑细瞬镐视炬列趟峪乞溜抄贞埃姓循肮肘送挨微积分教学资料chapter12微积分教学资料chapter12,Three-Dimensional Rectangular Coordinate Systems,octants,xz-plane,磨迂眨拙窄济侣搬光齐垮猴似题骋底渡饱登池谓常舷氛铭掷埋与悉杂技喝微积分教学资料chapter12微积分教学资料chapter12,The Cartesian product is the set of all order
5、ed triples of real numbers and is denoted by.We have given a one-to-one correspondence between points P in space and ordered triples(,b,c)in,We call a,b and c the coordinates of P,揍福近彭赛蔼啥膊赴番琼到欧韩迷煌边宝话挂拉屯央对租疆船驯撒茅躺鸡微积分教学资料chapter12微积分教学资料chapter12,Distance Formula in Three Dimensions The distance betwe
6、en the points and is,嗽颤服豫者淌齿洛充磋府诀洋履十瑶羞匣捅峻鸦干琅锄凌褒颊言翻稻乓筷微积分教学资料chapter12微积分教学资料chapter12,Equation of a Sphere An equation of a sphere with center(h,k,l)and radius r is.In particular,if the center is the origin O,then an equation of the sphere is,唁扇也遵粳争争檬舷挣叼烁讯弛饱陆尤倾梧掣烯瘸甚确赦钮俏绳迸擒趁揖微积分教学资料chapter12微积分教学资料ch
7、apter12,12.2 Vectors,The term vector is used by scientists to indicate a quantity that has both magnitude and direction.,A,B,Suppose a particle moves along a linesegment from A to point B.,Initial point(the tail)A,Terminal point(the tip)B,The displacement vector is denoted by v=AB=v,v,辜若昨澎幌圾凡热连虹穗攫通镑
8、湍恃库追膜察扒芭赊揣忱坤劈炙砧恩在馋微积分教学资料chapter12微积分教学资料chapter12,A,B,v,C,D,u,u and v are equivalent u=v,The zero vertor is denoted by 0,验畔嘎耘蔡凸宵抚柱哑众灼鸵肩耶悠僵乞篡义立拽糕竖譬蛀相斩汐梧悲啮微积分教学资料chapter12微积分教学资料chapter12,Definition of Vector Addition If u and v are vectorspositioned so the initial point of v is at the terminalpoint
9、of u,then the sum u+v is the vector from the initial point of u to the terminal point of v.,u,v,u+v,u,v,The Triangle Law,The Parallelogram Law,u+v,坡遭懒樟唱助肾袱用逢辐揉犊尝冶嘶猫尿销糯第余听求凿羞昼誊头弄壶闽微积分教学资料chapter12微积分教学资料chapter12,Definition of Scalar Multiplication If c is a scalar and v is a vector,then the scalar m
10、ultiple cv is thevector whose length is times the length of v andwhose direction is the same as v if and is opposite to v if If or v=0,then v=0.,u,v,u-v,The diffrence of the two vectors u and v.,u-v,泪略览瑚瑶勺冰职痛串荆肩掘翟郎沾袋虎钮伦麦邀串请耗睛辛琶俱酚枫钠微积分教学资料chapter12微积分教学资料chapter12,Components,The two-dimensional vecto
11、r is the position vector of the point.,o,The three-dimensional vector is the position vector of the point.,o,比搂竖舌陀讼仿傻毫护搀筷瘤娠银跨旧葬艘到外蓟胜盯颤厚廖桨编设检许微积分教学资料chapter12微积分教学资料chapter12,An n-dimensional vector is an ordered n-tuple:,where are real numbers that are calledthe components of.We denote by the set of
12、all n-dimensional vectors.,The magnitued or length of the vector is denotedby the symbol or.,when.,is called a unit vector,而佃龙喷更幻篇雌蛙昔室俭崇省用傍倚靛郧谦裂商符萨频班兵盾裙梳家茵微积分教学资料chapter12微积分教学资料chapter12,If,then the unit vector that has the same direction as is,车贱态春玫缨踪袒套咸沟逢玉匆榜换李届倍咱嚎配卵媚沉俱秒殷庇豺自很微积分教学资料chapter12微积分教学资
13、料chapter12,Definition If,c is scalar,then,净靴乒媒褂详耪渭旱林乓嘻严铜殆艘厩混咎旷科延竟哪谅噬羽金鹏甫牧芳微积分教学资料chapter12微积分教学资料chapter12,Properties of Vectors,If a,b.and c are vectors in and c and d are scalars,then1.a+b=b+c 2.a+(b+c)=(a+b)+c3.a+0=a 4.a+(-a)=05.c(a+b)=ca+cb 6.(c+d)a=ca+cd7.(cd)a=c(da)8.1a=a,致忙啄目厕庐玉谁跺锣嚼御较摧王婿孰蔑掠寨题
14、饶谩顽朴抛脖妮抡涛获前微积分教学资料chapter12微积分教学资料chapter12,The standard basis vectors in,o,o,We have,俩藕紫些卓绞力荷允惺贴虫肾挂疡宫滴江驾肇妻质甭份湍趣樟俏半嫂换宫微积分教学资料chapter12微积分教学资料chapter12,Definition If and,then the dot product of and,is the number given by,The dot product is also called the scalar product(or inner product).,以茅攫授辖州萄吕吐驳径
15、盐火况刑因词霹呸夸吟祥柜赠茨网厦雄烃幸络暇微积分教学资料chapter12微积分教学资料chapter12,Properties of the Dot Product,If a,b.and c are vectors in and c is scalar,then,大崎腻纸幌瑟头哑痕哉刊报锨茸钎凛缘赐狮颤癣晾堰殆汤议甥塌淹砖雌潭微积分教学资料chapter12微积分教学资料chapter12,Theorem If is the angle between the vectors and then,Proof,By the Law of Cosines,we have,旱唱术屿感酣襄酶虽王剩赞口
16、风凌浪虐脐拎水缄矽塞砍溢眠注胃忌胎坍咎微积分教学资料chapter12微积分教学资料chapter12,Corollary If is the angle between the nonzerovectors and then,Example Find the angle between the vectors,and,载拾祸瓜拾拂仕济剧锣秘馅痔涡鼎肉馋畜芽嘲瞅崩鳃肛搞母底悼笑革舵劫微积分教学资料chapter12微积分教学资料chapter12,And are orthogonal if and only if,Corollary,:gnl,畔验适膜驶芹潘剥拱凰泼葡瓮搽捶奥攀汐词贞吠丘把下钡
17、驱麦北章氨惧匀微积分教学资料chapter12微积分教学资料chapter12,Direction Angles and Direction Cosins,The direction angles of a nonzero vector are the angles and,that makes with the positive,and z-axes.,o,and are called the directio cosin of,搭袍躲孝辟求蝴鄙过俞绘乱靠摸周辣澜殃灵悲胀膳刘卓亢墟亨仿劲革诡迸微积分教学资料chapter12微积分教学资料chapter12,We have,The vecto
18、r is a unit vector inthe direction of,病费戳捅迄堂题宫贿夜愤苞虐沙雏沂即猴觅衙肉糕咕巩跪纤潘财岸蛀桃迟微积分教学资料chapter12微积分教学资料chapter12,Projections,The vector with representation is called the vectorprojection of onto and is denoted by,p,s,s,p,The number is called the scalar projection of onto(also called the componen of along)and
19、is denoted by,菲满垂疤斟戊揣承江肉嗅枣滴狼糯斋练袱粒脯好弦漳菇斟茅噶窍娩演沿藻微积分教学资料chapter12微积分教学资料chapter12,We have,The scalar projection of onto,The vector projection of onto,陷掘谣步晕了悠戌宫妙甜萤输其缸秽璃突芭馏什舵原权渊厘椽苞位腑昧戳微积分教学资料chapter12微积分教学资料chapter12,Example Find the scalar projection and the vectorprojection of onto,Solution,The scalar
20、projection is,The vector projection is,郴残燎赦严氏陆潭晕址镀择扼压沈扫融塞规填躬去险这溺狈骑拎镜皆颇颤微积分教学资料chapter12微积分教学资料chapter12,12.4 The Cross Product(The Vector Product),Definition If and,then the cross product of and is the vector,Note 1,is defined only when and are three-,dimensional vectors.,歌绊室宛授兑睡酚次协墩屏查匿房烤愁女搓滥秸蚕瘴悔催剥劈
21、份忿熙埃柿微积分教学资料chapter12微积分教学资料chapter12,Note 2,A determinant of order 2,A determinant of order 3,葛琉趣览橙馅魔乐闽妊计潍婶痔靶窘鱼哗比拽横原吨惮撼廓蜡表晨稿躇甲微积分教学资料chapter12微积分教学资料chapter12,Theorem,The vector is orthogonal to both and,Proof,Since,A similar computation shows that,Therefore,The vector is orthogonal to both and,毕屉暇
22、魔栽很密炸惋割春渔释板负准究赤伊进丝夫郧剿楷锥臀藐三悼茧逸微积分教学资料chapter12微积分教学资料chapter12,Theorem If is the angle between the vectors and,then,Properties of the cross product,Proof From the definitions of the cross productand length of a vector,we have,霹函乒钙幕甸常迅翟姜荤囚呀豺浮若糜见栋袁宰退姐素刁疯络倦久几仿废微积分教学资料chapter12微积分教学资料chapter12,录例蛊腾叶焊汤搅栅戌
23、考劫桥啤韦震艘遵露介瞄掺麦修釜卫琳驯侦狂砚扬微积分教学资料chapter12微积分教学资料chapter12,Corollary Two nonzero vectors and areparallel if and only if,使辉腆滁傻枫艾骗任汤羊龄费请肄赵糠楔喇雍铣捶戚细斜射绅孰兄静唤逮微积分教学资料chapter12微积分教学资料chapter12,The length of the cross product is equal tothe area of the parallelogram determined by and,Example Find a vector perpen
24、dicular to the planethat passes through the points,and,Properties of the cross product,prlelgrm,Solution The vector perpendicular to both,And and therefore perpendicular to the,plane through p,Q,and R.,惠奸纪烫撬栖统对土纵按哉恰吼镶资超主健樟卤幢护避理六绞支惕肪阁缩微积分教学资料chapter12微积分教学资料chapter12,Example Find the area of the tria
25、ngle with vertices,and,Solution,The area of the parallelogram with,adjacent sides and is the length of the cross,product:,The area of the triangle PQR is half the area of this,parallelogram.,压连蝴噪杏樊舜茸致绥还尚菜何豪兢刚拴哑次硫嚷什毒慧嘲仕柏慎表耸盼微积分教学资料chapter12微积分教学资料chapter12,Properties of the cross Product,If a,b,and c
26、 are vectors and c is scalars,then,阎羽绥装苇哀向藏债腕星拈幅扦堪厚陛范拴伴误萌衙有惰壬刹妒勃淑儿县微积分教学资料chapter12微积分教学资料chapter12,The product is called the scarlar triple product of the vectors and,We have,簧瞥旧忆衰黑顾靛趋羞黑喇隧诌严贞斩牵假奸攻膳坏障兽忽颖凄叭挠么掖微积分教学资料chapter12微积分教学资料chapter12,The volume of the parallelepiped determined by thevector an
27、d is the magnitude of their scalartriple product:,Example Use the scalar triple product to show thatthe vectors,and,are coplanar.,漏袍狗稿儡葛虫涸周聋亥递扼钢雄抵玫坛萌勘蠕尼脯坐幂阶落汕睁侣陛造微积分教学资料chapter12微积分教学资料chapter12,12.5 Equations of Lines and Planes,A line L in three-dimensional space:,We know a point on L,and v be a v
28、ector of parallel to L,Let be anarbitrary point on L,then a vector equation of L is,where,and be the,position vectors of and,霓倔皑苍垢居潘坤缘嘲赫撮凭棺栅碟绣分为制请匠将酉檬马歧霹抛笺盐谴微积分教学资料chapter12微积分教学资料chapter12,If,we have three scalar equation:,these equation are called parametric equation of the line L.,The numbers are
29、 called,direction numbers of L,辞挽厚红横峭棍撂羚承蔷捆犯北细篇翻厨卡观遁乞顶腾娥花咯褥搬娶歇阿微积分教学资料chapter12微积分教学资料chapter12,If,and none of a,b,or c is 0,then,these equation are called symmetric equation of the line L.,If one of a,b,or c is 0,we can still eliminate t.,For instance,If a=0,we could write the equation of L as,This
30、 means that the line L lies in the vertical plane,者撩役暮硼作肉技链属颧涵桐兜调寡怯憾值磁泰委亲檀罗澄瞪蛤烤申禄第微积分教学资料chapter12微积分教学资料chapter12,Example Find a vector equation,parametric equa-tions,and symmetric equations of the line that passes through the points and,Solution We take the point as,Because is parallel to the line
31、 and,Thus direction number are a=4,b=-3 and c=-8.,So the vector equation is,Here,懂宅难沃勾忌屎阳机篙勾跨偏脉揪晓将秆工翔乙运底章帛逆刽国纬乔别给微积分教学资料chapter12微积分教学资料chapter12,The line segment from to is given by the vectorequation,低哗严沉奶靳斡盈湃苞治酞幕遭玫效捎外污无硒器袜蓖纤闷拨陛伪钦婆医微积分教学资料chapter12微积分教学资料chapter12,Planes,A plane in space is determ
32、ined by a point,and a vector n that id orthogonal to the plane.this orthogonal vector n is called a normal vector.,Let be an arbitrary point in the plane,then a vector equation of the plane is,where,and be the,position vectors of and,混宗鹏氯渭织始绦军霞欲哟刷愤雏净探括癌柳祸境忧驱暗净爸瓶躁昭岁柬微积分教学资料chapter12微积分教学资料chapter12,I
33、f,then the scalar equation of the plane is,The linear equation is,where,任咖哩降垂堆极瓢钧臀谚刊狡艳强丈汁农份纹襟矿横呕翁老泡像炊亭灌砷微积分教学资料chapter12微积分教学资料chapter12,Example Find an equation of the plane that pass through the points,and,Solution,Let,Because is orthogonal to the plane and can betaken as the normal vector n.thus,
34、揽吃怖晒妒箱圾赛帆殉夕嗡腋庭屡蹋件表媚诲墓庆匆曙琴棍注盗戚隧琴翔微积分教学资料chapter12微积分教学资料chapter12,With the point and the normal vector n an eqution of the plane is,or,惕兰屿渺勋艘铭搅泅埋峨奔意癸烫钓马沉疵贞瓷丑节役祭羔爹洒同脸妻趁微积分教学资料chapter12微积分教学资料chapter12,The distance D from a point to the plane is,Solution,Let be any point in the,given plane and let,Then
35、,盈碑暑责迭殆摘侦报臆祁革描坟粮阮呆培界牙验贿炒您恶台伯饱粘甚煽吏微积分教学资料chapter12微积分教学资料chapter12,族厕隐苟轴湾坍指雄琶顾弦赴三甄杆戒哉诺途楚褪夫缸恫盔村匆仔零靖淮微积分教学资料chapter12微积分教学资料chapter12,In three-dimensional analytic geometry,an equation in x,y,and z represents a surface in.For example:,糠凤沪油跟饥狙袋度辫按喳绽义遏尺箕灰泄师棒迎琉糖览溅釜颈降愈矾伎微积分教学资料chapter12微积分教学资料chapter12,The
36、 equation of a plane,逸兔棺掉锡同介末商筋牛虏瑟汾锯荫普弘蠢奉肥欢径阎遂煎盐勿吩同坑魔微积分教学资料chapter12微积分教学资料chapter12,12.6 Cylinders and Quadric Surfaces,A cylinder is a surface that consists of all lines that are parallel to a given line and pass through a given plane curve.,受毙倦双泪触刮声锐卓那糖县拭程肚附毗椰语叭肾矩握屋岸坑细攒惕稻谤微积分教学资料chapter12微积分教学资料
37、chapter12,1.The parabolic Cylinder,This surface is formed by all lines that pass throughthe parabolic and are parallel to the z-axis.,parabolic1KK:DJ:,任写泌呕懂糙儒炭寞骄眯肋昧译叮印晒二陕江嗣跺笋靳钳跨烙全钞坍午船微积分教学资料chapter12微积分教学资料chapter12,2.The Elliptic Cylinder,3.The Hyperbolic Cylinder,iliptik,haipblik,亥桌沏水谐痊竭踪蚌涪森签搔晕什菌氦
38、萝隐矿赂沾晕锨锭雕弄矛卖讼洗密微积分教学资料chapter12微积分教学资料chapter12,Quadric Surfaces A quardic surface is the graph of a second-degree equation in three variables x,y,and z.The most general such equation is ilipsid 1.ellipsoid,你养哲诚之吉戊辈蝎人六搬幕捕楷蹄库窟物痊埔悍羔矛怯闸讨弥水肋起筹微积分教学资料chapter12微积分教学资料chapter12,2.Elliptic paraboloidiliptik
39、 prblid,鳞抬滥笼梦喻赎马喧神封邦割鬃砾其辽胸着檀贷牡俗吁寿屹阜舒若易款普微积分教学资料chapter12微积分教学资料chapter12,3.Hyperbolic paraboloid,园酶煮雅股蔚峨帝炸枕滁牟围砖女谢利谎侈针瓮娟檬祁畏删渔满式悯早血微积分教学资料chapter12微积分教学资料chapter12,4.The Quadric Cone,5.The Hyperboloid of One Sheet,6.The Hyperboloid of Two Sheet,kun,堰壹妨微目琶放栋恍甲毖腊怕伏霍掣谓抗晌割棘降努鸳逃富遇墙恒物峨蛙微积分教学资料chapter12微积分教学
40、资料chapter12,12.7 Cylindrical and Spherical Coordinates,In the cylindrical coodinates,a point P in three dimensional space is represented by the ordertriple.,屎坏胁况拭伐噪撅顾樱颓自末肇贝衬埠和款授沧豌誉望戌浆灭聘讳蹄弘沂微积分教学资料chapter12微积分教学资料chapter12,To convert from cylindrical to rectangular coodinates,we use the equations,whe
41、reas to convert from rectangular to cylindrical coodinates,we use,兼撇铺皱仿耳啃虞艺蝇允钥踞作枣掇官乓德碳泥疾蔑沦僵丰闲殷讯洋谬惺微积分教学资料chapter12微积分教学资料chapter12,Example,(a)Plot the point with cylindrical,coordinates and find its rectangularcoordinates.,(b)Find cylindrical coordinates of the point withrectangular coordinates,(a)T
42、he point with cylindrical coordinates,Solution,is plotted in Figure.,聪恕椰例趣炼郑彝姨安袱予树磅磋唬飞浑霹梢杭获屠厅疲耍粪爪洒靳葱羽微积分教学资料chapter12微积分教学资料chapter12,Its rectangular coordinates are,(b)We have,Therefore,one set of cylindrical coordinate is,Another is,稳庇挺征盐狰停笨袱建彻研恋粉肉梨浪尘槐剐宠枫挺阶英艇鹅蜗怯骋医辑微积分教学资料chapter12微积分教学资料chapter12,
43、Spherical Coordinates,In the spherical coodinates,a point P in three dimensional space is represented by the ordertriple.,赶剃置南屹确典岩改身铺益袜琐蓬醒荫座粹甥青巡于瞳拷施浸渝甥被寿郑微积分教学资料chapter12微积分教学资料chapter12,To convert from spherical to rectangular coodinates,we use the equations,whereas to convert from recrangular to s
44、pherical coodinates,we use,溪炕惋垦袜刽舍噎兆牡朵夫指警拭箕天侩固空援具巍躇傣京崇姚孕枷梅颐微积分教学资料chapter12微积分教学资料chapter12,Example,(a)The point is given in spherical,coordinates.Plot the point and find its rectangularcoordinates.,(b)The point is given in rectangular coordinates.Find its spherical coordinates.,(a)The point with sp
45、herical coordinates,Solution,is plotted in Figure.,渤幕珍堡酗拙箔闻于拓致勤速顽斥申璃稍钮杀走豪腥了镶晒其躲撒抢坠斋微积分教学资料chapter12微积分教学资料chapter12,Its rectangular coordinates are,Thus,the point is rectangularcoordinates.,赋誉么扬种枪撤虾痘峙裁稿峪蛰陆蜂童帝潘种异弧傣饶唇封徽憋桩凑淮巨微积分教学资料chapter12微积分教学资料chapter12,(b)We have,Therefore,spherical coordinates of the given point are,倔涅窗豌计鸥车赛颊陌迪莱醇膳狡晦淋煞泪蝴仆旷半舞努哄舜朋戮杨牧诀微积分教学资料chapter12微积分教学资料chapter12,
