Transcription of Lecture 3: Coordinate Systems and Transformations
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Lecture 3: Coordinate Systems and Transformations
Lecture3:CoordinateSystemsandTransformat ionsTopics:1. Coordinatesystemsandframes2. Changeof frames3. A netransformations4. Rotation,translation,scaling,andshear5. RotationaboutanarbitraryaxisChapter4, , , , , , vectorv2lR3canbe representedas a linearcombinationofthreelinearlyindepend ent basisvectorsv1,v2,v3,v= 1v1+ 2v2+ 3v3:Thescalars 1, 2, 3arethecoordinatesofv. We typicallychoosev1= (1;0;0),v2= (0;1;0),v3= (0;0;1) .v2v1v3 1v = 1v1 + 2v2 + 3v32 Supposewe want to (linearly)changethebasisvectorsv1,v2,v3t ou1,u2,u3. We expressthenewbasisvectorsas combinationsof theoldones,u1=a11v1+a12v2+a13v3;u2=a21v1 +a22v2+a23v3;u3=a31v1+a32v2+a33v3;andthu s obtaina 3 3 `changeof basis'matrixM=0@a11a12a13a21a22a23a31a32 a331A:If thetwo representationsof a givenvectorvarev=aT0@v1v2v31A;andv=bT0@u 1u2u31A;wherea= ( 1 2 3)Tandb= ( 1 2 3)T, thenaT0@v1v2v31A=v=bT0@u1u2u31A=bTM0@v1v 2v31A;which impliesthata=MTbandb= (MT) 1a:3 This3 Dcoordinatesystemis not,h
Coordinate systems and frames Recall that a vector v 2 lR3 can be represented as a linear combination of three linearly independent basis vectors v1, v2, v3, v = 1v1 + 2v2 + 3v3: The scalars 1, 2, 3 are the coordinates of v. We typically choose v1 = (1;0;0), v2 = (0;1;0), v3 = (0;0;1) . v2 v1 v3 α1 v = α1v1 + α2v2 + α3v3 2
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