Chapter3 - Chapter 3 Vector Geometry MAT188H1F Lec0106...

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Chapter 3: Vector GeometryMAT188H1F Lec0106 BurbullaChapter 3 Lecture NotesFall 2010Chapter 3 Lecture NotesMAT188H1F Lec0106 BurbullaChapter 3: Vector GeometryChapter 3: Vector Geometry3.1: Geometric Vectors3.2: The Dot Product and Projections3.5: The Cross Product3.3: Lines and Planes in 3 Dimensions3.4: Matrix Transformations ofR2Chapter 3 Lecture NotesMAT188H1F Lec0106 Burbulla
Chapter 3: Vector Geometry3.1: Geometric Vectors3.2: The Dot Product and Projections3.5: The Cross Product3.3: Lines and Planes in 3 Dimensions3.4: Matrix Transformations ofR2IntroductionUp until now we have been emphasizing algebra, and ignoringgeometry. But there are geometric properties associated with1.column or row matrices, which are called vectors,2.matrix multiplication of vectors by square matricies, whichrepresent transformations.In Chapter 3 we will look specifically at vectors in the plane, and inspace; and at transformations of the plane. The vectors we willlook at in Sections 3.1, 3.2 and 3.5 are the same as the vectorsthat you may already be familiar with from physics: velocity,acceleration, force, momentum, angular momentum, etc. Section3.3 will look at lines and planes in space. Section 3.4 will covertransformations of the plane. Some, or all of this, may have beencovered in your high school. Section 3.5 summarizes key resultsabout the cross product of vectors inR3.Chapter 3 Lecture NotesMAT188H1F Lec0106 BurbullaChapter 3: Vector Geometry3.1: Geometric Vectors3.2: The Dot Product and Projections3.5: The Cross Product3.3: Lines and Planes in 3 Dimensions3.4: Matrix Transformations ofR2Vectors as Directed Line SegmentsVectors can be represented by directed line segments. Vectors havemagnitude and direction, but are not located in any one particularplace. The magnitude of a vector is represented by the length of aline segment. The direction of a vector is indicated by an arrow.The direction of a vector can also be specified by an angle withrespect to thex-axis, or even with respect to they-axis.vuThe diagram to the left illustratestwo vectors,uandv. Since theyhave the same length,and thesame direction, we sayu=v.Chapter 3 Lecture NotesMAT188H1F Lec0106 Burbulla
Chapter 3: Vector Geometry3.1: Geometric Vectors3.2: The Dot Product and Projections3.5: The Cross Product3.3: Lines and Planes in 3 Dimensions3.4: Matrix Transformations ofR2The Length of a VectorThe length of a vectorvis denoted byv.For a vectorrepresented as a directed line segment, the length is simply thelength of the line segment. The length of a vector is non-negative.The only vector which has length zero is the zero vector,represented by 0.The zero vector, which can be considered as asingle point, is also unique in the sense that it is the one vectorwhich does not have a well-defined direction.

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