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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 CoordinatesIn this chapter we introduce vectors andcoordin
2、ate 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 vectors provide particularly simple descriptions of lines and planes in
3、space.12.1 Three-Dimensional Coordinate Systems Coordinate axesxyzoxyzoCoordinate planesplanexyxz-planeplaneyz origin OThrough point O,three axes vertical each other,by right-hand rule,we obtain a Three-Dimensional Rectangular Coordinate SystemsxyzThree-Dimensional Rectangular Coordinate Systemso oc
4、tantsplanexyplaneyzxz-plane The Cartesian product is the set of all ordered 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,|),(RzyxzyxRRR3R3Rxyzo)0,0,(aP),0,0(cR)0,(baA)0,0(bQ),(coaC),0(cbB),(cbaMrWe call a,b
5、 and c the coordinates of PDistance Formula in Three Dimensions The distance between the points and is 21PP),(1111zyxP),(2222zyxP21221221221)()()(zzyyxxPP1z2zO2xxz1xABC1P2P1y2yyEquation 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
6、,then an equation of the sphere is2222)()()(rlzkyhxoxyz2222rzyx12.2 VectorsThe term vector is used by scientists to indicate a quantity that has both magnitude and direction.ABSuppose a particle moves along a linesegment from A to point B.Initial point(the tail)ATerminal point(the tip)BThe displacem
7、ent vector is denoted by v=AB=vvABvCDuu and v are equivalent u=vThe zero vertor is denoted by 0Definition of Vector Addition If u and v are vectorspositioned so the initial point of v is at the terminalpoint of u,then the sum u+v is the vector from the initial point of u to the terminal point of v.u
8、vu+vuvThe Triangle LawThe Parallelogram Lawu+vDefinition of Scalar Multiplication If c is a scalar and v is a vector,then the scalar multiple 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.c0c.0c0ccuvu-vThe diff
9、rence of the two vectors u and v.u-vComponentsThe two-dimensional vector is the position vector of the point .21,aaa),(21aaP),(21aaPopoaThe three-dimensional vector is the position vector of the point .321,aaaa),(321aaaP),(321aaaPopoaAn n-dimensional vector is an ordered n-tuple:naaaa,21where are re
10、al numbers that are calledthe components of .We denote by the set ofall n-dimensional vectors.naaa,21aThe magnitued or length of the vector is denotedby the symbol or .aaa22221naaaanvwhen .1aais called a unit vectorIf ,then the unit vector that has the same direction as is0aaaauDefinition If,21naaaa
11、nbbbb,21c is scalar,thennnnnbabababbbaaaba,22112121nncacacaaaacac,2121Properties of VectorsIf 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=anvThe standard basis vectors in ,0,0,1i3v,0,1,0j1,0,0koijkoi
12、jk321,aaaakajaiaaaaa321321,We have Definition If and naaaa,21nbbbb,21,then the dot product of and is the number given bybaabnnbabababa2211The dot product is also called the scalar product(or inner product).Properties of the Dot ProductIf a,b.and c are vectors in and c is scalar,thennv0a0 5.b)(ab)(ab
13、a)(4.cabac)(ba 3.abba.2aaa.12cccTheorem If is the angle between the vectors and thenab0cosbabaProofBy the Law of Cosines,we have cos2222bababaCorollary If is the angle between the nonzerovectors and thenabbabacosExample Find the angle between the vectors kjia 2andkjib23And are orthogonal if and only
14、 ifab0baCorollary:gnl Direction Angles and Direction CosinsThe direction angles of a nonzero vector are the angles and ,that makes with the positive a,a,yxand z-axes.321,aaaao,0,cos,cosand are called the directio cosin of cosaWe have aaaaaa321cos,cos,coscos,cos,cosThe vector is a unit vector inthe d
15、irection of aProjectionsThe vector with representation is called the vectorprojection of onto and is denoted by.projababpspsbasbapThe number is called the scalar projection of onto (also called the componen of along )and is denoted by pabWe have The scalar projection of ontob:abaabababbbcoscompaThe
16、vector projection of ontob:aaabaaabaab2a)(projExample Find the scalar projection and the vectorprojection of onto.2kjiakjib23SolutionThe scalar projection isababacompThe vector projection isaabbaacompproj12.4 The Cross Product(The Vector Product)Definition If and 321,aaaa,21nbbbbthen the cross product of and is the vector ab122131132332,babababababababaNote 1is defined only when and are three-abdimensional vectors.kbbaajbbaaibbaabbbaaakjiba212131313232321321Note 2A determinant of order 222112121
