e) Whatis the area of the final imagecircle?If is a polygonmatrixfor a set of pointsin a coordinateplane,we coulduse matrixarithmeticto find
, the matrixof the imageof the polygonafter the translation-dilationof this example4.
Let’s use this translation-dilationto movethe rectanglein example3.
Dilationscalaris
TranslationmatrixisThe final imageis the rectanglewith verticesat and.
LessonSummary
In this lessonwe completedour studyof transformations.Dilationscompletethe collectionof transformations
we havenow learnedabout:translations,reflections,rotations,and dilations.
Scalarmultiplicationwas defined.Differencesof scalarmultiplicationcomparedto matrixmultiplicationwere
observed:any scalarcan multiplyany matrix,and the dimensionsof a scalarproductare the sameas the
dimensionsof the matrixbeingmultiplied.
Compositionsinvolvingdilationsgaveus anotherway to changeand movepolygons.All sortsof matrix
operations—scalarmultiplication,matrixmultiplication,and matrixaddition—canbe usedto find the image
of a polygonin thesecompositions.
Pointsto Consider
All of our workwith the matricesthat representpolygonsand translationsin two-dimensionalspace(a coor-
dinateplane)has ratherobviousparallelsin threedimensions.
A matrixthat represents pointswouldhave rowsand columnsratherthan.
A dilationis still a scalarproduct.