# Chapter3 - Chapter 3 Vector Geometry MAT188H1F Lec0106...

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Chapter 3: Vector Geometry MAT188H1F Lec0106 Burbulla Chapter 3 Lecture Notes Fall 2010 Chapter 3 Lecture Notes MAT188H1F Lec0106 Burbulla Chapter 3: Vector Geometry Chapter 3: Vector Geometry 3.1: Geometric Vectors 3.2: The Dot Product and Projections 3.5: The Cross Product 3.3: Lines and Planes in 3 Dimensions 3.4: Matrix Transformations of R 2 Chapter 3 Lecture Notes MAT188H1F Lec0106 Burbulla

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Chapter 3: Vector Geometry 3.1: Geometric Vectors 3.2: The Dot Product and Projections 3.5: The Cross Product 3.3: Lines and Planes in 3 Dimensions 3.4: Matrix Transformations of R 2 Introduction Up until now we have been emphasizing algebra, and ignoring geometry. But there are geometric properties associated with 1. column or row matrices, which are called vectors, 2. matrix multiplication of vectors by square matricies, which represent transformations. In Chapter 3 we will look speciﬁcally at vectors in the plane, and in space; and at transformations of the plane. The vectors we will look at in Sections 3.1, 3.2 and 3.5 are the same as the vectors that you may already be familiar with from physics: velocity, acceleration, force, momentum, angular momentum, etc. Section 3.3 will look at lines and planes in space. Section 3.4 will cover transformations of the plane. Some, or all of this, may have been covered in your high school. Section 3.5 summarizes key results about the cross product of vectors in R 3 . Chapter 3 Lecture Notes MAT188H1F Lec0106 Burbulla Chapter 3: Vector Geometry 3.1: Geometric Vectors 3.2: The Dot Product and Projections 3.5: The Cross Product 3.3: Lines and Planes in 3 Dimensions 3.4: Matrix Transformations of R 2 Vectors as Directed Line Segments Vectors can be represented by directed line segments. Vectors have magnitude and direction, but are not located in any one particular place. The magnitude of a vector is represented by the length of a line segment. The direction of a vector is indicated by an arrow. The direction of a vector can also be speciﬁed by an angle with respect to the x -axis, or even with respect to the y -axis. ± ± ± ± > ~ v ± ± ± ± > ~ u The diagram to the left illustrates two vectors, ~ u and ~ v . Since they have the same length, and the same direction, we say ~ u = ~ v . Chapter 3 Lecture Notes MAT188H1F Lec0106 Burbulla
Chapter 3: Vector Geometry 3.1: Geometric Vectors 3.2: The Dot Product and Projections 3.5: The Cross Product 3.3: Lines and Planes in 3 Dimensions 3.4: Matrix Transformations of R 2 The Length of a Vector The length of a vector ~ v is denoted by k ~ v k . For a vector represented as a directed line segment, the length is simply the length of the line segment. The length of a vector is non-negative.

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## This note was uploaded on 12/23/2010 for the course MAT mat 188 taught by Professor Dietrichburbulla during the Spring '10 term at University of Toronto.

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Chapter3 - Chapter 3 Vector Geometry MAT188H1F Lec0106...

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