高數(shù)英文版課件

1、Lecture 5Planes and Lines in Space2Equations for Planes in SpaceA plane in space is determinedby knowing a point on the planeand its “tilt” or orientation.Then M is the sets of all pointsSuppose that plane M passesnonzero vectorand is normal (perpendicular) to thethrough a pointfor whichis orthogona

2、l to n.Thus, the dot productThis equation is equivalent toorPoint normal form scalar equation of the plane3Equations for Planes in SpaceEquation for a planehasThe plane throughnormal toComponent equation simplified:Vector equation:Component equation:whereTwo planes are parallel if and only if their

3、normals are parallel, orfor some scalar k.4Finding an Equation for a planeperpendicular toFind an equation for the plane throughSolution The component equation isSimplifying, we obtain5Finding an Equation for a Plane through Three PointsFind an equation for the plane throughandSolution I We have to

4、find a vector normal to theplane and use it with one of the points(it does not matter which) to write anequation for the plane.The cross product6Finding an Equation for a Plane Through Three PointsSolution (continued)It is easy to see that n is normal to the plane.We substitute the components of thi

5、s vectorintoand the coordinate of the point-normal form of the equationand obtain7Solution II is any point in the plane,Suppose thatSince these three vector are coplanarif and only if the point P lies in the plane,thenso we haveFinding an Equation for a Plane Through Three Points8Solution (continued

6、)that is Expanded the determinant on the leftside of above equation, we haveFinding an Equation for a Plane Through Three Points9Intercept Form of the Equation for a PlaneIn general, if the intercepts of the plane with the x-axis, y-axis and z-axisfor the planeandbeThen, just as the lastrespectively

7、.example, we can obtain the equationIntercept form of the equation10General Equations for PlanesGeneral Equation for a planeThe equation can be rewritten in the formwhereTherefore, the equation of any plane is a linear equation in threeConversely, any linear equation in three variables representsvar

8、iables.if A, B, C are not all 0.a plane with normal vectorthe equation can be written asIn fact, if11Some Planes with Special Locations(1) If a given plane passes through the originthensatisfy the general equation for the plane, so thatTherefore, the equation of the plane through the origin is12Some

9、 Planes with Special Locations(2) If a given plane is parallel to the z-axis,andTherefore, the equation of this plane isis orthogonal to Similarly, the equations of planes whichare parallel to the x-axis or y-axis arerespectively.then the normal vector13Some Planes with Special Locations(3) If a giv

10、en plane is orthogonal to the z-axis, Therefore, the equation of this plane isSimilarly, the equations of planes whichare orthogonal to the x-axis or y-axis arerespectively.and so orthen14Some Problems about Planes(1) Angle Between PlanesThe angle between two intersectingplanes is defined to be the

11、(acute)angle determined by the normalvectors as shown in the figure.Let andThere normal vectors can be chosen asThen respectively.15If two planes are parallel or orthogonal, their normal vector also parallel or orthogonal.Some Problems about Planes(2) Position Relationships Between Two Planes16We ch

12、oose arbitrarily a given point which does not lie in the plane p.be a given plane,Let and draw a pointin the plane p a vector point to the plane p is equal to the absolutevalue of the projection of the vectorThen the distance from aonto the normal vector n of p.Thus, by the formula of projection, we

13、 getbeand Some Problems about Planes (3) Distance From a Point to a Plane17The Distance from a Point to a PlaneSince ThereforeThus the formula of the distance isandlies in the plane p,Notice thatso that18Some Examples about PlanesExample 1.Discuss the position relationships between the followingplan

14、es: Solution (1)Since then these two planes intersect and the angle between them is19Solution (2)Since andthen these two planes are parallel.Again, sincebutthese two planes are not same.Example 1.Discuss the position relationships between the followingplanes: Some Examples about Planes20Example 1.Di

15、scuss the position relationships between the followingplanes: Solution (3)Since these two planes are parallel. Again, sinceandthese two planes aresame.Some Examples about Planes21Example 2.Find an equation for the plane p that passes throughand is parallel to the planethe pointSolution Let the norma

16、l vector to the plane p be n; thencan be taken as the normal vectorand soorThus the equation of the plane p isof p.Some Examples about Planes22Since the two point P1 and P2 lie in the plane, we haveExample 3.Find an equation for the plane p that passes throughThe two pointsand is perpendicular to th

17、e planeSolution (I)be the equation for the plane p.LetandBecause p is perpendicular to the planewe haveThen we have Therefore, the equation for p isSome Examples about Planes23Example 3.Find an equation for the plane p that passes throughThe two pointsand is perpendicular to the planeThus, the equat

18、ion of the plane p isSolution (II)Let the normal vector to the plane p be n.Then where by the given conditions, thenAlso,Some Examples about Planes24Review (Planes)Equations for planes in spacePoint-normal form of the equationPlane through three pointsIntercept form of the equationGeneral equation f

19、or a planeSome equations of the planes with special locations Some problems about planesAngle between planesPosition relationship between two planeDistance from a point to a plane25Equations for Lines in SpaceIn the plane, a line is determined by a point and a number giving theAnalogously, in space

20、a line is determined by a pointslope of the line.and a vector giving the direction of the line.Suppose that L is a line in space passingparallel through a pointThen Lto a vectoris the set of all pointsfor whichis parallel to v.26and this last equation can be rewrittenEquations for Lines in SpaceThe

21、value of t depends onasthe location of the point P along the line, and the domain of t is Thus for some scalar parameter t.isThe expanded form of the equation(1)27form for the equation of a line in space.on the line andis the position vector of a pointIfis thethen we have the following vectorpositio

22、n vector of pointVector Equation for a LineA vector equation for the line L throughparallel to v isare the position vectors of andandpoint on the line, respectively.Equations for Lines in Space28Equations for Lines in Space(1)These equations give us the standard parametrization of the line for thepa

23、rameter intervalEquating the corresponding components of the two sides of Equation(1) Gives three scalar equations involving the parameter t :Parametric Equation for a Lineparallel The standard parametrization of the line throughisto(2)29Parametrizing a Line Through a Point Parallel to a Vectorparal

24、lel toFind parametric equation for the line throughSolution equation (2) becomeWith equalequalandtoto(2)30Parametrizing a Line Through Two PointsandFind parametric equations for the line throughSolution The vectoris parallel to the line,as the “base point” and writtenWe could have chosenThese equati

25、ons serve as well as the first;different point on the line for a given value of t .giveand equation (2) withthey simply place you at a(2)31If we eliminate the parameter t in the equations, we obtain theequivalent formscalled the symmetric form equation ( point-direction form equation ) of L.Equation

26、s for Lines in Spacelies on the line L if and only if theObviously, a point satisfy the equationscoordinatesofIn this case, we writeIfthis impliesorthe direction of L32Lines of IntersectionTwo planes that are not parallel intersect at a line.Suppose that the equations of two planes areandThen the eq

27、uation for the line of intersectioncan be represented by the systemof equationsThis is called the generalform of the equation of the line.33Finding a Vector Parallel to the Line of Intersection of Two PlanesFind a vector parallel to the line of intersection of the planesandSolution As in the right f

28、igure,the required vector is34Parametrizing the Line of Intersection of Two PlanesFind parametric equations for the line in which the planesintersect.andSolution The last example identifiesas a vector parallel to the line.To find a point on the line, we can takeany point common to the two planes.sub

29、stitutingin the planeequations and solving for x and y simultaneously identifies one of thesepoints asThe line is(2)35Finding the Equation for a PlaneFind the equation of a plane p that passes through the line L ofandintersection of the two planesand is perpendicular to the plane p2.Solution (I)It i

30、s easy to see thatis on L and the directionvector of L isBy theassumptions, we know Notice thatthenn can be chosen as36Finding the Equation for a PlaneSolution (I) (continued)Sincelies in the plane p , the equation of p isor37The Method of Pencil of PlanesLet the equation of L bewhere the parameter

31、t is an arbitrary real constant,Then the equationorplanes through line L except the planeThe equation of the pencil of planes through line Lrepresents all 38Finding the Equation for a PlaneFind the equation of a plane p that passes through the line L ofandintersection of the two planesand is perpend

32、icular to the plane p2.Solution (II)The equation of the pencil of planesthrough L isSince p is perpendicular to the plane p2,we have39Finding the Equation for a PlaneSolution (II) (continued)Solving this equation, we obtain thatand therefore the equation of p isCompare this result with the previousr

33、esult, it is easy to see that this methodserves just as the previous method.40Some Problems about Lines(1) Angle between two linesThe included angle between two lines is defined as the acute anglebetween their direction vectors.Thus, two lines are parallel (ororthogonal) iff their direction vectors

34、are parallel (or orthogonal).41Let andbe two given lines.Then their direction vectors can be chosen asrespectively.andBy the formula ofIncluded angle between two vectors, we haveBy this formula, we can easily find the angle between two lines.Some Problems about Lines(1) Angle between two lines42Then

35、, by the necessary and sufficient conditions for two vectors to beparallel and orthogonal, we obtain:Some Problems about Lines(1) Angle between two linesQuestion: Do two lines intersect if they are neither parallel nor orthogonal ? 43Position Relationships Between Two LinesDiscuss the position relat

36、ionships between the two linesIf they intersect, find the point of intersection. If they are coplanar, findthe plane p in which they lie.Solution The direction vectors of L1 and L2 areandandrespectively.Then the cosine of the angle between these two line isThen these two lines neither parallel nor o

37、rthogonal.44Position Relationships Between Two LinesSolution (continued)It is easy to see that pointisis onon the lineandshould be coplanar.andSinceare coplanar, the vectorThen, if andTherefore, these two line intersect.Are they coplanar?are coplanar.Thus andDo they intersect?45Position Relationship

38、s Between Two LinesSolution (continued)we haveTherefore, the point of intersection of these two linesParametrizing andandSolving the equationswe findisWhere is the point of intersection?46Position Relationships Between Two LinesSolution (continued)Since are coplanar, we will find the equation of the

39、 plane.andThen the equation of the plane isorThe normal vector of the plane isWhat is the equation of the plane?47The included angle between the line Land the plane p is defined as the acuteangle j between L and its projectionvector in the plane p.be a plane and LetThen the included angle betweenbe a line.of L and the normal vector n of p is the direction vectororSome Problems about Lines(2) Angle between a line