2433-Ch12-slides

2433-Ch12-slides - 3-D Coordinate System:(12.1)P: (x1,...

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Unformatted text preview: 3-D Coordinate System:(12.1)P: (x1, y1,z1) andQ: (x2, y2, z2),points in 3-space:(A)Distance Formula:d(P, Q) =(x2-x1)2+ (y2-y1)2+ (z2-z1)2(B)Midpoint Formula:The mid-pointRof the line segment joiningPandQis the pointR:x1+x22,y1+y22,z1+z22.1(C)Equation for the sphere ofradiusrand centerP(a, b, c):(x-a2+ (y-b)2+ (z-c)2=r2.Examples:1.The pointsA: (2,-1,3)andB: (-4,5,-1)are the endpoints of a diameter ofa sphere. Find an equation for thesphere.22.The pointsA: (2,-2,1), B: (1,1,3), C: (2,,5)are the vertices of a right triangle.Find an equation of the sphere withcenter at the midpoint of the hy-potenuse and passing through thevertex opposite the hypotenuse.Vectors:(12.2, 12.3)Avectorinn-dimensional space isa directed line segment; it is repre-sented by an orderedn-tuple of realnumbers.3-Space:A vectorain 3-spaceis an ordered triple of numbers:a= (a1, a2,a3)The vector= (0,,0) is thezerovector.3Operations on VectorsLeta= (a1,a2, a3) andb= (b1, b2,b3)be vectors in 3-space and letbea real number (scalar).(A)Equality:a=biffa1=b1, a2=b2, a3=b3.4(B)Vector Addition:a+b= (a1+b1, a2+b2, a3+b3).Motivation from physics:5Properties of Addition:Leta,b,cbe vectors.1.a+b=b+a(commutative)2.(a+b) +c=a+ (b+c)(associative)3.a+=+a=a,is theadditive identity64.to each vectorathere corre-sponds a uniquexsuch thata+x=x+a=(additive inverse)xis denoted-aSubtraction:a-b=a+ (-b) = (a1-b1, a2-b2, a3-b3)7(C)Multiplication by a Scalar:a= (a1, a2, a3).Motivation from Physics:Properties of Mult. by a Scalar:1.1a=a,a=2.(+)a=a+a,(a+b) =a+bdistributivelaws83.()a=(a) =(a)NOTES:1.aandbareparalleliffa=bfor some number.2.is parallel to every vector;= 0afor alla.(D)Norm (Magnitude) & Di-rection:Thenormofa, denoted bya,is:a=a21+a22+a23,= 0.ais a nonnegative number; it isthelength of the vectora.9Properties of Norm:1.a0;a= 0 iffa=.2.a=||a.3.a+ba+b.(triangleinequality)10Unit Vectors:uis aunit vectorifu= 1Ifbis a non-zero vector, thenub=1bbis a unit vector in the same directionasb.11Unit Coordinate Vectors:i= (1,,0),j= (0,1,0),k= (0,,1)i,j,k-Representation:a= (a1, a2, a3) =a1i+a2j+a3kDirection:????has no direction.12Dot Product:(12.4)Leta= (a1,a2, a3) andb= (b1, b2,b3)be vectors. Thedot productofaandb, denotedabis the numberab=a1b1+a2b2+a3b3....
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2433-Ch12-slides - 3-D Coordinate System:(12.1)P: (x1,...

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