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

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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.2
2.The pointsA: (2,-2,1), B: (1,1,3), C: (2,0,5)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 vector0= (0,0,0) is thezerovector.3
Operations on VectorsLeta= (a1, a2, a3) andb= (b1, b2, b3)be vectors in 3-space and letαbea 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:5
Properties of Addition:Leta,b,cbe vectors.1.a+b=b+a(commutative)2.(a+b) +c=a+ (b+c)(associative)3.a+0=0+a=a,0is theadditive identity6
4.to each vectorathere corre-sponds a uniquexsuch thata+x=x+a=0(additive inverse)xis denoted-a
Subtraction:a-b=a+ (-b) = (a1-b1, a2-b2, a3-b3)7
(C)Multiplication by a Scalar:αa= (α a1, α a2, α a3).Motivation from Physics:Properties of Mult. by a Scalar:1.1a=a,0a=02.(α+β)a=αa+βa,α(a+b) =αa+αbdistributivelaws8
3.(αβ)a=α(βa) =β(αa)NOTES:1.aandbareparalleliffa=λbfor some numberλ.2.0is parallel to every vector;0= 0afor alla.
(D)Norm(Magnitude)&Di-rection:Thenormofa,denoted bya,is:a=a21+a22+a23,0= 0.ais a nonnegative number; it isthelength of the vectora.9
Properties of Norm:1.a0;a= 0 iffa=0.2.αa=|α|a.3.a+ba+b.(triangleinequality)10
Unit Vectors:uis aunit vectorifu= 1Ifbis a non-zero vector, thenub=1bbis a unit vector in the same directionasb.11
Unit Coordinate Vectors:i= (1,0,0),j= (0,1,0),k= (0,0,1)i,j,k-Representation:a= (a1, a2, a3) =a1i+a2j+a3kDirection:????0has no direction.12
Dot Product:(12.4)Leta= (a1, a2, a3) andb= (b1, b2, b3)

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