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1 1. Vectors

2 1.1 Vectors and Matrices Linear algebra is concerned with two basic kinds of quantities: vectors and matrices.

3 1.1 Vectors and Matrices Scalars and Vectors - Scalar: a numerical value denoted by lowercase italic type such as a, k, v, w, and x ex) temperature, length, and speed - Vector: a numerical value and a direction denoted by lowercase boldface type such as a, k, v, w, and x ex) velocity, force, and displacement

4 1.1 Vectors and Matrices Scalars and Vectors Terminal point Initial point Magnitude Direction

5 1.1 Vectors and Matrices Equivalent Vectors Bound vector Free vector In this text we will focus exclusively on free vectors, leaving the study of bound vectors for courses in engineering and physics. Two vectors v and w are equal (also called equivalent) if they are represented by parallel arrows with the same length and direction. v=w. The vector whose initial and terminal points coincide has length zero, so we call this zero vector and denote it by 0.

6 1.1 Vectors and Matrices Vector Addition

7 1.1 Vectors and Matrices Vector Addition

8 1.1 Vectors and Matrices Vector Subtraction

9 1.1 Vectors and Matrices Scalar Multiplication

10 1.1 Vectors and Matrices Vectors in Coordinate Systems Rectangular coordinate system in 2-space y-axis origin x-axis One-to-one correspondence between points in the plane and ordered pairs of real numbers P point a, b coordinates P a, b

11 1.1 Vectors and Matrices Vectors in Coordinate Systems Rectangular coordinate system in 3-space x-axis, y-axis, z-axis Left-handed Right-handed In this text we will work exclusively with right-handed coordinate systems.

12 1.1 Vectors and Matrices Vectors in Coordinate Systems If a vector v in 2-space or 3-space is positioned with its initial point at the origin of a rectangular coordinate system, then the vector is completely determined by the coordinates of its terminal point, and we call these coordinates the components of v relative to the coordinate system. The set of all vectors in 2-space: R 2 The set of all vectors in 3-space: R 3

13 1.1 Vectors and Matrices Components of a Vector Whose Initial Point Is Not At The Origin v is a vector in R 2 with initial point P 1 (x 1,y 1 ) and terminal point P 2 (x 2,y 2 ). v P P OP OP x x, y y The components of v are obtained by subtracting the coordinates of the initial point from the corresponding coordinates of the terminal point.

14 1.1 Vectors and Matrices Vectors in R n You can think of the numbers in an n-tuple (v 1, v 2,, v n ) as either the coordinates of a generalized point or the components of a generalized vector. 0=(0,0,0,,0) We will call this the zero vector or sometimes the origin of R n.

15 1.1 Vectors and Matrices Equality of Vectors

16 1.1 Vectors and Matrices Equality of Vectors

17 1.1 Vectors and Matrices Sums of Three or More Vectors

18 1.1 Vectors and Matrices Parallel and Collinear Vectors

19 1.1 Vectors and Matrices Linear Combinations

20 1.1 Vectors and Matrices Application to Computer Color Models Colors on computer monitors are commonly based on what is called the RGB color model. Each color vector c in RGB space or the RGB color cube is expressible as a linear combination of the form

21 1.1 Vectors and Matrices Alternative Notations for Vectors Comma-delimited form Row-vector form Column-vector form

22 1.1 Vectors and Matrices Matrices We define a matrix to be a rectangular array of numbers, called the entries of the matrix. You can also think of a matrix as a list of row vectors or column vectors.

23 1.2 Dot Product and Orthogonality Norm of A Vector The length of a vector v in R 2 and R 3 is commonly denoted by the symbol v. From the theorem of Pythagoras,

24 1.2 Dot Product and Orthogonality Norm of A Vector

25 1.2 Dot Product and Orthogonality Unit Vectors A vector of length 1 is called a unit vector. Normalizing v: Example 2 Find the unit vector u that has the same direction as v=(2,2,-1).

26 1.2 Dot Product and Orthogonality Standard Unit Vectors When a rectangular coordinate system is introduced in R 2 or R 3, the unit vectors in the positive directions of the coordinate axes are called the standard unit vectors. In R 2 these vectors are denoted by In R 3 these vectors are denoted by

27 1.2 Dot Product and Orthogonality Standard Unit Vectors Standard unit vectors in R n

28 1.2 Dot Product and Orthogonality Distance between Points in R n

29 1.2 Dot Product and Orthogonality Dot Products Example 3 International Standard Book Number or ISBN

30 1.2 Dot Product and Orthogonality Algebraic Properties of The Dot Products

31 1.2 Dot Product and Orthogonality Algebraic Properties of The Dot Products Example 4

32 1.2 Dot Product and Orthogonality Angle Between Vectors in R 2 and R 3 The angle between u and v: the smallest nonnegative angle θ through which one of the vectors can be rotated in the plane of the vectors until it coincides with the other. Algebraically, the radian measure of is in the interval 0 θ π, and in R 2 the angle is generated by a counterclockwise rotation.

33 1.2 Dot Product and Orthogonality Angle Between Vectors in R 2 and R 3

34 1.2 Dot Product and Orthogonality Angle Between Vectors in R 2 and R 3 Example 5 Find the angle θ between a diagonal of a cube and one of its edges.

35 1.2 Dot Product and Orthogonality Angle Between Vectors in R 2 and R 3 Example 6

36 1.2 Dot Product and Orthogonality Orthogonality Example 7 Example 8 If 0 is the zero vector in R n, then 0 v=0. Thus 0 is orthogonal to R n. w v=0 for every vector v in R n w=0

37 1.2 Dot Product and Orthogonality Orthonormal Sets

38 1.2 Dot Product and Orthogonality Euclidean Geometry in R n

39 1.2 Dot Product and Orthogonality Euclidean Geometry in R n

40 1.2 Dot Product and Orthogonality Euclidean Geometry in R n

41 1.2 Dot Product and Orthogonality Euclidean Geometry in R n

42 1.2 Dot Product and Orthogonality Euclidean Geometry in R n

43 1.3 Vector Equations of Lines and Planes Vector And Parametric Equations of Lines Example 1

44 1.3 Vector Equations of Lines and Planes Lines Through Two Points Example 1

45 1.3 Vector Equations of Lines and Planes Point-Normal Equations of Planes Example 3

46 1.3 Vector Equations of Lines and Planes Vector and Parametric Equations of Planes

47 1.3 Vector Equations of Lines and Planes Vector and Parametric Equations of Planes Example 4 Example 5 A plane is uniquely determined by three noncollinear points. Find the plane that passes through the points P(2,-4,5), Q(-1,4,-3), and R(1,10,-7).

48 1.3 Vector Equations of Lines and Planes Vector and Parametric Equations of Planes Example 6 Find a vector equation of the plane whose parametric equations are Example 7 Find parametric equations of the plane x-y+2z=5.

49 1.3 Vector Equations of Lines and Planes Lines and Planes in R n

50 1.3 Vector Equations of Lines and Planes Comments on Terminology A vector v lies on a line L in R 2 and R 3 if the terminal point of the vector lies on the line when the vector is positioned with its initial point at the origin.

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