01. Vectors 20130305 - Yonsei Universityweb.yonsei.ac.kr/hgjung/Lectures/MAT203/01 Vectors.pdf ·...

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1 Vectors1 Vectors


11 Vectors and Matrices11 Vectors and Matrices

Linear algebra is concerned with two basic kinds of quantities ldquovectorsrdquo and ldquomatricesrdquo



- 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

Scalars and Vectors



Initial point

Terminal point

MagnitudeDirection

Scalars and Vectors



Bound vector Free vector

Equivalent Vectors

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



Vector Addition



Vector Addition



Vector Subtraction



Scalar Multiplication



Vectors in Coordinate Systems

Rectangular coordinate system in 2-space

x-axis

y-axis

origin

One-to-one correspondence between points in the plane and ordered pairs of real numbers

P a b

point coordinates P a b





x-axis y-axis z-axis

Left-handed Right-handed

In this text we will work exclusively with right-handed 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 R2




Components of a Vector Whose Initial Point Is Not At The Origin

v is a vector in R2 with initial point P1(x1y1) and terminal point P2(x2y2)

1 2 2 1 2 1 2 1PP OP OP x x y y v

The components of v are obtained by subtracting the coordinates of the initial point from the corresponding coordinates of the terminal point



Vectors in Rn

You can think of the numbers in an n-tuple (v1 v2 hellip vn) as either the coordinates of a generalized point or the components of a generalized vector

0=(000hellip0)

We will call this the zero vector or sometimes the origin of Rn



Equality of Vectors



Equality of Vectors



Sums of Three or More Vectors



Parallel and Collinear Vectors



Linear Combinations



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



Alternative Notations for Vectors

Comma-delimited form

Row-vector form

Column-vector form



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


12 Dot Product and Orthogonality12 Dot Product and Orthogonality

Norm of A Vector

The length of a vector v in R2 and R3 is commonly denoted by the symbol ||v||

From the theorem of Pythagoras



Norm of A Vector



Unit Vectors

A vector of length 1 is called a unit vector

Normalizing v

Example 2Example 2

Find the unit vector u that has the same direction as v=(22-1)



Standard Unit Vectors

When a rectangular coordinate system is introduced in R2 or R3 the unit vectors in the positive directions of the coordinate axes are called the standard unit vectors

In R2 these vectors are denoted by





Standard unit vectors in Rn



Distance between Points in Rn



Dot Products

Example 3Example 3

International Standard Book Number or ISBN

0-471-15307-9



Algebraic Properties of The Dot Products




Example 4Example 4



Angle Between Vectors in R2 and R3

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 0leθleπ and in R2 the angle is generated by a counterclockwise rotation







Example 5Example 5Find the angle θ between a diagonal of a cube and one of its edges




Example 6Example 6



Orthogonality

Example 7Example 7

If 0 is the zero vector in Rn then 0∙v=0 Thus 0 is orthogonal to Rn

wv=0 for every vector v in Rn w=0

Example 8Example 8



Orthonormal Sets



Euclidean Geometry in Rn














13 Vector Equations of Lines and Planes13 Vector Equations of Lines and Planes

Vector And Parametric Equations of Lines

Example 1Example 1


Lines Through Two Points


Example 1Example 1


Point-Normal Equations of Planes


Example 3Example 3


Vector and Parametric Equations of Planes





Example 4Example 4

Example 5Example 5A plane is uniquely determined by three noncollinear points

Find the plane that passes through the points P(2-45) Q(-14-3) and R(110-7)




Example 6Example 6Find a vector equation of the plane whose parametric equations are

Example 7Example 7

Find parametric equations of the plane x-y+2z=5


Lines and Planes in Rn



Comments on Terminology


A vector v lies on a line L in R2 and R3 if the terminal point of the vector lies on the line when the vector is positioned with its initial point at the origin



Linear algebra is concerned with two basic kinds of quantities ldquovectorsrdquo and ldquomatricesrdquo









Scalars and Vectors



Initial point

Terminal point

MagnitudeDirection

Scalars and Vectors




Equivalent Vectors






Vector Addition



Vector Addition



Vector Subtraction








x-axis

y-axis

origin


P a b



















1 2 2 1 2 1 2 1PP OP OP x x y y v




Vectors in Rn


0=(000hellip0)




Equality of Vectors



Equality of Vectors









Linear Combinations










Row-vector form

Column-vector form



Matrices





Norm of A Vector





Norm of A Vector



Unit Vectors


Normalizing v

Example 2Example 2

















Dot Products

Example 3Example 3


0-471-15307-9







Example 4Example 4















Example 6Example 6



Orthogonality

Example 7Example 7



Example 8Example 8



Orthonormal Sets



















Example 1Example 1




Example 1Example 1




Example 3Example 3







Example 4Example 4







Example 7Example 7

















Scalars and Vectors



Initial point

Terminal point

MagnitudeDirection

Scalars and Vectors




Equivalent Vectors






Vector Addition



Vector Addition



Vector Subtraction








x-axis

y-axis

origin


P a b



















1 2 2 1 2 1 2 1PP OP OP x x y y v




Vectors in Rn


0=(000hellip0)




Equality of Vectors



Equality of Vectors









Linear Combinations










Row-vector form

Column-vector form



Matrices





Norm of A Vector





Norm of A Vector



Unit Vectors


Normalizing v

Example 2Example 2

















Dot Products

Example 3Example 3


0-471-15307-9







Example 4Example 4















Example 6Example 6



Orthogonality

Example 7Example 7



Example 8Example 8



Orthonormal Sets



















Example 1Example 1




Example 1Example 1




Example 3Example 3







Example 4Example 4







Example 7Example 7











Initial point

Terminal point

MagnitudeDirection

Scalars and Vectors




Equivalent Vectors






Vector Addition



Vector Addition



Vector Subtraction








x-axis

y-axis

origin


P a b



















1 2 2 1 2 1 2 1PP OP OP x x y y v




Vectors in Rn


0=(000hellip0)




Equality of Vectors



Equality of Vectors









Linear Combinations










Row-vector form

Column-vector form



Matrices





Norm of A Vector





Norm of A Vector



Unit Vectors


Normalizing v

Example 2Example 2

















Dot Products

Example 3Example 3


0-471-15307-9







Example 4Example 4















Example 6Example 6



Orthogonality

Example 7Example 7



Example 8Example 8



Orthonormal Sets



















Example 1Example 1




Example 1Example 1




Example 3Example 3







Example 4Example 4







Example 7Example 7












Equivalent Vectors






Vector Addition



Vector Addition



Vector Subtraction








x-axis

y-axis

origin


P a b



















1 2 2 1 2 1 2 1PP OP OP x x y y v




Vectors in Rn


0=(000hellip0)




Equality of Vectors



Equality of Vectors









Linear Combinations










Row-vector form

Column-vector form



Matrices





Norm of A Vector





Norm of A Vector



Unit Vectors


Normalizing v

Example 2Example 2

















Dot Products

Example 3Example 3


0-471-15307-9







Example 4Example 4















Example 6Example 6



Orthogonality

Example 7Example 7



Example 8Example 8



Orthonormal Sets



















Example 1Example 1




Example 1Example 1




Example 3Example 3







Example 4Example 4







Example 7Example 7











Vector Addition



Vector Addition



Vector Subtraction








x-axis

y-axis

origin


P a b



















1 2 2 1 2 1 2 1PP OP OP x x y y v




Vectors in Rn


0=(000hellip0)




Equality of Vectors



Equality of Vectors









Linear Combinations










Row-vector form

Column-vector form



Matrices





Norm of A Vector





Norm of A Vector



Unit Vectors


Normalizing v

Example 2Example 2

















Dot Products

Example 3Example 3


0-471-15307-9







Example 4Example 4















Example 6Example 6



Orthogonality

Example 7Example 7



Example 8Example 8



Orthonormal Sets



















Example 1Example 1




Example 1Example 1




Example 3Example 3







Example 4Example 4







Example 7Example 7











Vector Addition



Vector Subtraction








x-axis

y-axis

origin


P a b



















1 2 2 1 2 1 2 1PP OP OP x x y y v




Vectors in Rn


0=(000hellip0)




Equality of Vectors



Equality of Vectors









Linear Combinations










Row-vector form

Column-vector form



Matrices





Norm of A Vector





Norm of A Vector



Unit Vectors


Normalizing v

Example 2Example 2

















Dot Products

Example 3Example 3


0-471-15307-9







Example 4Example 4















Example 6Example 6



Orthogonality

Example 7Example 7



Example 8Example 8



Orthonormal Sets



















Example 1Example 1




Example 1Example 1




Example 3Example 3







Example 4Example 4







Example 7Example 7











Vector Subtraction








x-axis

y-axis

origin


P a b



















1 2 2 1 2 1 2 1PP OP OP x x y y v




Vectors in Rn


0=(000hellip0)




Equality of Vectors



Equality of Vectors









Linear Combinations










Row-vector form

Column-vector form



Matrices





Norm of A Vector





Norm of A Vector



Unit Vectors


Normalizing v

Example 2Example 2

















Dot Products

Example 3Example 3


0-471-15307-9







Example 4Example 4















Example 6Example 6



Orthogonality

Example 7Example 7



Example 8Example 8



Orthonormal Sets



















Example 1Example 1




Example 1Example 1




Example 3Example 3







Example 4Example 4







Example 7Example 7
















x-axis

y-axis

origin


P a b



















1 2 2 1 2 1 2 1PP OP OP x x y y v




Vectors in Rn


0=(000hellip0)




Equality of Vectors



Equality of Vectors









Linear Combinations










Row-vector form

Column-vector form



Matrices





Norm of A Vector





Norm of A Vector



Unit Vectors


Normalizing v

Example 2Example 2

















Dot Products

Example 3Example 3


0-471-15307-9







Example 4Example 4















Example 6Example 6



Orthogonality

Example 7Example 7



Example 8Example 8



Orthonormal Sets



















Example 1Example 1




Example 1Example 1




Example 3Example 3







Example 4Example 4







Example 7Example 7


























1 2 2 1 2 1 2 1PP OP OP x x y y v




Vectors in Rn


0=(000hellip0)




Equality of Vectors



Equality of Vectors









Linear Combinations










Row-vector form

Column-vector form



Matrices





Norm of A Vector





Norm of A Vector



Unit Vectors


Normalizing v

Example 2Example 2

















Dot Products

Example 3Example 3


0-471-15307-9







Example 4Example 4















Example 6Example 6



Orthogonality

Example 7Example 7



Example 8Example 8



Orthonormal Sets



















Example 1Example 1




Example 1Example 1




Example 3Example 3







Example 4Example 4







Example 7Example 7











Vectors in Rn


0=(000hellip0)




Equality of Vectors



Equality of Vectors









Linear Combinations










Row-vector form

Column-vector form



Matrices





Norm of A Vector





Norm of A Vector



Unit Vectors


Normalizing v

Example 2Example 2

















Dot Products

Example 3Example 3


0-471-15307-9







Example 4Example 4















Example 6Example 6



Orthogonality

Example 7Example 7



Example 8Example 8



Orthonormal Sets



















Example 1Example 1




Example 1Example 1




Example 3Example 3







Example 4Example 4







Example 7Example 7











Equality of Vectors



Equality of Vectors









Linear Combinations










Row-vector form

Column-vector form



Matrices





Norm of A Vector





Norm of A Vector



Unit Vectors


Normalizing v

Example 2Example 2

















Dot Products

Example 3Example 3


0-471-15307-9







Example 4Example 4















Example 6Example 6



Orthogonality

Example 7Example 7



Example 8Example 8



Orthonormal Sets



















Example 1Example 1




Example 1Example 1




Example 3Example 3







Example 4Example 4







Example 7Example 7











Equality of Vectors









Linear Combinations










Row-vector form

Column-vector form



Matrices





Norm of A Vector





Norm of A Vector



Unit Vectors


Normalizing v

Example 2Example 2

















Dot Products

Example 3Example 3


0-471-15307-9







Example 4Example 4















Example 6Example 6



Orthogonality

Example 7Example 7



Example 8Example 8



Orthonormal Sets



















Example 1Example 1




Example 1Example 1




Example 3Example 3







Example 4Example 4







Example 7Example 7

















Linear Combinations










Row-vector form

Column-vector form



Matrices





Norm of A Vector





Norm of A Vector



Unit Vectors


Normalizing v

Example 2Example 2

















Dot Products

Example 3Example 3


0-471-15307-9







Example 4Example 4















Example 6Example 6



Orthogonality

Example 7Example 7



Example 8Example 8



Orthonormal Sets



















Example 1Example 1




Example 1Example 1




Example 3Example 3







Example 4Example 4







Example 7Example 7


















Row-vector form

Column-vector form



Matrices





Norm of A Vector





Norm of A Vector



Unit Vectors


Normalizing v

Example 2Example 2

















Dot Products

Example 3Example 3


0-471-15307-9







Example 4Example 4















Example 6Example 6



Orthogonality

Example 7Example 7



Example 8Example 8



Orthonormal Sets



















Example 1Example 1




Example 1Example 1




Example 3Example 3







Example 4Example 4







Example 7Example 7











Matrices





Norm of A Vector





Norm of A Vector



Unit Vectors


Normalizing v

Example 2Example 2

















Dot Products

Example 3Example 3


0-471-15307-9







Example 4Example 4















Example 6Example 6



Orthogonality

Example 7Example 7



Example 8Example 8



Orthonormal Sets



















Example 1Example 1




Example 1Example 1




Example 3Example 3







Example 4Example 4







Example 7Example 7











Norm of A Vector





Norm of A Vector



Unit Vectors


Normalizing v

Example 2Example 2

















Dot Products

Example 3Example 3


0-471-15307-9







Example 4Example 4















Example 6Example 6



Orthogonality

Example 7Example 7



Example 8Example 8



Orthonormal Sets



















Example 1Example 1




Example 1Example 1




Example 3Example 3







Example 4Example 4







Example 7Example 7











Norm of A Vector



Unit Vectors


Normalizing v

Example 2Example 2

















Dot Products

Example 3Example 3


0-471-15307-9







Example 4Example 4















Example 6Example 6



Orthogonality

Example 7Example 7



Example 8Example 8



Orthonormal Sets



















Example 1Example 1




Example 1Example 1




Example 3Example 3







Example 4Example 4







Example 7Example 7











Unit Vectors


Normalizing v

Example 2Example 2

















Dot Products

Example 3Example 3


0-471-15307-9







Example 4Example 4















Example 6Example 6



Orthogonality

Example 7Example 7



Example 8Example 8



Orthonormal Sets



















Example 1Example 1




Example 1Example 1




Example 3Example 3







Example 4Example 4







Example 7Example 7
























Dot Products

Example 3Example 3


0-471-15307-9







Example 4Example 4















Example 6Example 6



Orthogonality

Example 7Example 7



Example 8Example 8



Orthonormal Sets



















Example 1Example 1




Example 1Example 1




Example 3Example 3







Example 4Example 4







Example 7Example 7















Example 4Example 4















Example 6Example 6



Orthogonality

Example 7Example 7



Example 8Example 8



Orthonormal Sets



















Example 1Example 1




Example 1Example 1




Example 3Example 3







Example 4Example 4







Example 7Example 7























Example 6Example 6



Orthogonality

Example 7Example 7



Example 8Example 8



Orthonormal Sets



















Example 1Example 1




Example 1Example 1




Example 3Example 3







Example 4Example 4







Example 7Example 7











Orthogonality

Example 7Example 7



Example 8Example 8



Orthonormal Sets



















Example 1Example 1




Example 1Example 1




Example 3Example 3







Example 4Example 4







Example 7Example 7











Orthonormal Sets



















Example 1Example 1




Example 1Example 1




Example 3Example 3







Example 4Example 4







Example 7Example 7



























Example 1Example 1




Example 1Example 1




Example 3Example 3







Example 4Example 4







Example 7Example 7












Example 1Example 1




Example 3Example 3







Example 4Example 4







Example 7Example 7












Example 3Example 3







Example 4Example 4







Example 7Example 7















Example 4Example 4







Example 7Example 7













Example 7Example 7









01. Vectors 20130305 - Yonsei Universityweb.yonsei.ac.kr/hgjung/Lectures/MAT203/01 Vectors.pdf ·...

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