Chapter 3 Vectors in 2-Space and 3-Space

Chapter Contents

3.1 Introduction to Vectors (Geometric) 3.2 Norm of a Vector; Vector Arithmetic 3.3 Dot Product; Projections 3.4 Cross Product 3.5 Lines and Planes in 3-Space

3.1 Introduction to Vectors (Geometric)

Geometric Vectors Symbolically, we shall denote vectors in lowercase

boldface type. All our scalars will be real numbers and will be denoted in lowercase italic type

initial point

terminal point • The vector of length zero is called the zero vector and is denoted by 0. • Since there is no natural direction for the zero vector • the negative of v, is defined to be the vector having the same magnitude as v, but oppositely directed.

Definition If v and w are any two

vectors, then the sum v+w is the vector determined as follows: Position the vector w so that its initial point coincides with the terminal point of v. The vector v+w is represented by the arrow from the initial point of v to the terminal point of w.

Definition

If v and w are any two vectors, then the difference of w from v is defined by v – w = v + (-w)

Definition If v is a nonzero vector

and k is nonzero real number (scalar), then the product kv is defined to be the vector whose length is |k| times the length of v and whose direction is the same as that of v if k > 0 and opposite to that of v if k < 0. We define kv =0 if k = 0 or v = 0.

A vector of the form kv is called a scalar multiple.

Vectors in coordinate Systems(1/2)

In Figure 3.1.6, that v has been positioned so its initial point is at the origin of a rectangular coordinate system. The coordinates of the terminal point of v are called the components of v, and we write ),( 21 vvv

21,vv

Vectors in coordinate Systems(2/2)

If two vectors are

equivalent if and only if and

),( and ),( 2121 wwwvvv

2211 and wvwv

Vectors in 3-Space (1/4)

rectangular coordinate system

origin

coordinate axes• Each pair of coordinate axes determines a plane called a coordinate plane. These are referred to as the xy- plane, the xz-plane, and the yz-plane.

• To each point P in 3- space we assign a triple of numbers (x, y, z), called the coordinates of P.

Vectors in 3-Space (2/4)

Rectangular coordinate systems in 3-space fall into two categories, left-handed and right-handed.

In this book we shall use only right-handed coordinate systems.

Vectors in 3-Space (3/4) A vector v in 3-space is

positioned so its initial point is at the origin of a rectangular coordinate system. The coordinates of the terminal point of v are called the components of v, and we write

),,( 321 vvvv

),,( and ),,( 321321 wwwwvvvv If are two vectors in 3-space, then

scalarany is where),,,(

),,(

,, vifonly and if equivalent are and

321

332211

332211

kkvkvkvkv

wvwvwvwv

wvwvwwv

Vectors in 3-Space (4/4)

),(

is )(

point terminaland )(point initialth vector wi thespace-2In

),,(),,(),,(

then ,),,(point terminaland

),,(point initial has vector theIf

origin. at thenot ispoint initial its

that so positioned is vector a Sometimes

121221

111

111

12121211122221

2222

111121

yyxxPP

yxP

yxP

zzyyxxzyxzyxPP

zyxP

zyxPPP

Example 1Vector Computations with Components

If v=(1,-3,2) and w=(4,2,1),then

v + w=(5,-1,3), 2v=(2,-6,4) -w=(-4,-2,-1),

v – w=v + (-w)=(-3,-5,1)

Example 2Finding the components of a Vector

)12,6,5()4)8(),1(5,27(

are )8,5,7(point terminaland

)4,1,2(point initial with vector theof components The

2

121

v

P

PPPv

Translation of Axes In Figure 3.1.14a we have translated

the axes of an xy-coordinate system to obtain an x’y’-coordinate system whose O’ is at point (x ,y)=(k ,l ).

A point P in 2-space now has both (x ,y) coordinates and (x’ ,y’) coordinates.

x’= x – k , y’= y – l , these formulas are called the translation equations.

In 3-space the translation equations are x’= x – k , y’= y – l , z’= z – m where ( k, l, m ) are the xyz-coordinates of the x’y’z’-origin.

Example 3Using the Translation Equations (1/2)

Suppose that an xy-coordinate system is translated to obtain an x’y’-coordante system whose origin has xy-coordinates (k ,l )=(4,1).

(a) Find the x’y’-coordinate of the point with the xy-coordinates P(2,0) (b) Find the xy-coordinate of the point with the x’y’-coordinates Q(-1,5)

Solution (a). The translation equations are x’=x-4, y’=y-1 so the x’y’-coordinate of P(2,0) are x’=2-4=-2 and y’=0-1=-1. Solution (b). The translation equations in (a) can be

written as x=x’+4, y=y’+1 so the xy-coordinate of Q are x=-1+4=3 and y=5+1=6.

Example 3Using the Translation Equations (2/2)

3.2 Norm of a Vector; Vector Arithmetic

Theorem 3.2.1Properties of Vector Arithmetic

If u, v and w are vectors in 2- or 3-space and k and l are scalars, then the following relationship

Norm of a Vector (1/2) The length of a vector u is often called

the norm of u and is denoted by .

Figure (a): it follows from the Theorem of Pythagoras that the norm of a vector

in 2-space is

Figure (b): Let be a vector in 3-space.

A vector of norm 1 is called a unit vector.

u

),( 21 uuu 22

21 uuu

23

22

21 uuuu

),,( 321 uuuu

Norm of a Vector (2/2) If

are two points in 3-space, then the distance s between them is the norm of vector

Similarly in 2-space:

the length of the vector ku :

),,( and ),,( 22221111 zyxPzyxP

),,(

12121221

21

zzyyxxPP

PP

ukku

Example 1Finding Norm and Distance

11244)51()13()24(

is )1,3,4(P and (2,-1,5)P points ebetwwen th d distance The

14)1()2()3(u

is ) (-3,2,1u vector theof norm The

222

21

222

d

3.3 Dot Product; Projections

The Angle Between Vectors Let u and v be two nonzero vectors in 2-

space or 3-space, and assume these vectors have been positioned so their initial points coincided. By the angle between u and v, we shall mean the angleθ determined by u and v that satisfies 0 ≤ θ ≤ π.

Component Form of the Dot Product (1/2)

)( or

)( cos

as (2) rewritecan we, Since

(2) cos 2-

yields cosines of law then the

v,andu between angle theis 3.3.3, figurein shown as If

vectors.nonzero twobe ),,( and ),,(Let

222

21

222

21

222

321321

uvvuvu

uvvuvu

uvPQ

vuvuPQ

vvvvuuuu

Component Form of the Dot Product (2/2)

2211

332211

233

222

211

2

23

22

21

223

22

21

2

: space-2in Similarly

Simplfyingafter obtain we

)()()( and

,

ngSubstituti

vuvuvu

vuvuvuvu

uvuvuvuv

vvvvuuuu

The formula is also valid if u=0 or v=0.

Finding the Angle Between Vectors If u and v are nonzero vectors then

it also can be written as

vu

vu

vuvu

cos

(1) cos

Example 2Dot Product Using [3]

Example 4Finding Dot products from Components

Orthogonal Vectors Perpendicular vectors are also called

orthogonal vectors. In light of Theorem 3.l.1b, two nonzero

vectors are orthogonal if and only if their dot product is zero.

To indicate that u and v are orthogonal vectors we write u ⊥v.

Example 5A Vector Perpendicular to a Line Show that in 2-space the nonzero vector

n=(a,b) is perpendicular to the line ax+by+cz=0.

Solution

lar.perpendicu are and Thus,

0or 0),(),(

form in the expressed becan which

0)()(

obtain we(6)in equations the

gsubtractinon But lar.perpendicu are andn that showonly need we

3.3.5), (Figure line thealong runs ),( vector theSince

(6) 0

0

thatso line, on the pointsdistinct be ),( and ),(Let

21

211212

1212

21

121221

22

11

222111

PPn

PPnyyxxba

yybxxa

PP

yyxxPP

cbyax

cbyax

yxPyxP

Theorem 3.3.2Properties of the Dot Product

If u, v and w are vectors in 2- or 3-space and k is a scalar, then:

An Orthogonal Projection (1/2)

To "decompose" a vector u into a sum of two terms, one parallel to a specified nonzero vector a and the other perpendicular to a.

Figure 3.3.6: Drop a perpendicular from the tip of u to the line through a, and construct the vector w1 from Q.

Next form the difference:uwuwwwwuw )( then 112112

An Orthogonal Projection (2/2)

The vector w1 is called the orthogonal projection of u on a or sometimes the vector component of u along a. It is denoted by

The vector w1 is called the vector component of u orthogonal to a. Since we have , this vector can be written in notation (7) as

(7) uproja

uprojuw a 2

12 wuw

Example 6Vector Component of u Along a

zero. isproduct dot their

that showingby lar perpendicu are and vector t theVerify tha

),,(),,()3,1,2(

is a toorthogonalu ofcomponent vector theand

),,()2,1,4(

is a alongu ofcomponent vector theThus,

212)1(4

15)2)(3()1)(1()4)(2(

a. toorthogonalu ofcomponent vector theand

a alongu ofcomponent vector theFind .)2,1,4( and )3,1,2(Let

711

72

76

710

75

720

710

75

720

2115

2

2222

auproju

uproju

aa

auuproj

a

au

au

a

a

a

Solution :

Example 7Distance Between a Point and a Line (1/2)

221010010100

0

0

0

11

000

n ),()( ),,( But

thus,n;on QP of projection

orthogonal theoflength the toequal is D distance thefigure, in the indicated As

3.3.8). (Fig line thelar toperpendicu isn vector theExample5, of By virtue

.at ispoint

initial its that so ),(n vector heposition t and line on thepoint any be ),(Let

.0 line theand ),(Ppoint between D distance for the formula a Find

bayybxxanQPyyxxQP

n

nQPQPprojD

Q

bayxQ

cbyaxyx

n

Solution :

Example 7Distance Between a Point and a Line (2/2)

(13)

formula theyields (12)in expression thisngSubstituti

or 0

so line, theofequation the

satisfy scoordinate its line, on the lies )y,(point theSince

(12) )()(

that so

22

00

1111

11

22

1010

ba

cbyaxD

byaxccbyax

xQba

yybxxaD

Solution (count)

Example 8Using the Distance Formula

5

11

25

11

43

6)2(4)1)(3(

is 06-4y3x line theto

(1,-2)point thefrom D distance that the(13) Formula from followsIt

22

D

3.4 Cross Product

Cross Product of Vectors Recall from Section 3.3 that the dot

product of two vectors in 2-space or 3-space produces a scalar.

We will now define a type of vector multiplication that produces a vector as the product, but which is applicable only in 3-space.

Example 1Calculating a Cross Product

Example 2u×v Is Perpendicular to u and to v

Determinant Form of Cross Product (1/2) A cross product can be represented

symbolically in the form of 3 × 3 determinant:

For example :

kji

kji

vu 672

103

221

then(3,0,1), vand (1,2,-2)u if

Example 5Calculating a Scalar Triple Product (1/2)

Independence of Cross Product and Coordinates (2/2) Question: two fixed vectors u and v might have

different cross products in different coordinate systems. Recall :

Since these properties of u x v depend only on the lengths and relative positions of u and v and not on the particular right-hand coordinate system being used.

Thus, we say that the definition of u x v is coordinate free.

This result is of importance to physicists and engineers who often work with many coordinate systems in the same problem.

the direction

the length

3.5 Lines and Planes in 3-Space

Planes in 3-Space One can specify a plane in 3-space by giving its

inclination and specifying one of its points. A convenient method for a plane is to specify a

nonzero vector, called a normal, that is perpendicular to the plane.

We want to find the equation of a plane passing through the point ; and have a nonzero vector n = (a. b. c) as a normal. From Figure 3.5.1:

We call this the point-normal form of the equation

of a plane.

),,( 0000 zyxP

0)()()(

0

is, that n, toorthogonal is vector the

000

0

0

zzcyybxxa

PPn

PP

Example 1Finding the Point-Normal Equation of a Plane

The Solution of a System in 3-Space

planes. threeof

on intersecti of points

the tocorrespond

system a ofsolution the

3

2

1

kizhygx

kfzeydx

kczbyax

Example 2Equation of a Plane Through Three Points (1/2)

Example 2Equation of a Plane Through Three Points (2/2)

0)(

so ,r-rThen plane. the tonormal vector a

be c)b,(a,nlet and )z,y,(xPpoint theto

origin thefrom vector thebe ),,(let

0

00

0000

0000

rrn

PP

zyxr

Vector Form of Equation of a Plane Referring to Figure 3.5.3, let r=( x, y, z) be the

vector from the origin to the point P (x , y, z),

This is called the vector form of the equation of a plane.

Example 3Vector Equation of a Plane Using(5)

Lines in 3-space

(6)

such thatdalar t a is therefor which is,that

v, toparallel is vector thefor which

),,( points thoseofprecisely consists

that 3.5.4) (Figureclear isIt ).,,(vector

nonzero the toparallel and ),,(point the

through space-3 line theis that Suppose space.-3in

linesfor equationsobtain tohow show now shall We

0

0

0000

tvPP

PP

zyxP

lcbav

zyxP

l

Example 4Parametric Equations of a Line

Example 5Intersection of a Line and the xy-Plane

Example 6Line of Intersection of Two Planes

Vector form of Equation of a Line

Example 7A Line Parallel to a Given Vector

Example 8Distance Between a Pont and a Plane

Example 9Distance Between Parallel Planes