Chapter 3 Vectors[1]

8/8/2019 Chapter 3 Vectors[1]

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Vectors

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CHAPTER 3

BASIC CONCEPTS

What is scalar?

- a quantity that has only magnitude

What is vector?

- a quantity that has magnitude and direction

A vector can be represented by a directedline segment where the

i) length of the line segment represents

the magnitude of the vector

ii) direction of the line segment represents

the direction of the vector

VECTORS


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A vector can be written as QP

, or . The

order of the letters is important. QP

means

the vector is from P to Q, QP means vectoris from Q to P.

IfP 11 , yx is the initial point andQ 22 ,yx is the terminal point of a directedline segment, PQ, then component form of

vector v that represents PQ is

12

12

12121,,

yy

xxyyxxvv x

and the magnitude or the length ofv is

2

2

2

1

2

12

2

12

cc

yyxx

v

P

Q

Initial point

Terminal PointPQ


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Example:

Find the component form and length of the

vector v that has initial point 7,3 and

terminal point 5,2 .

Example:

Given 5,2v and 4,3w , find each of the

following vectors.

a) v21 b) vw c) wv 2


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Any vector that has magnitude of 1 unit iscalled unit vector.

Theorem

Ifa is a non-null vector and if a is a unit

vector having the same direction as a, then

a

aa

Example:

Find a unit vector in the direction of5,2

v and verify that it has length 1.


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STANDARD UNIT VECTORS

Three standard unit vectors are: i,j dan k

Vectors i,j and k can be written in

components form:

i = ,j = < 0,y, 0 >, k = < 0, 0,z >

and can interpreted as

a =

=xi +yj +zk


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The vector QP

with initial point

111 ,, zyxP and terminal point 222 ,, zyxQ has the standard representation

ki )()()( 121212 zzyyxxQP or

121212 ,, zzyyxxPQ

Example:

Let u be the vector with initial point (2,-5) and

terminal point (-1,3), and let jiv 2 . Writeeach of the following vectors as a linear

combination ofi andj.

a) u

b) vuw 32


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Example:

The vector v has a length of 3 and makes an

angle of 630

with the positivex-axis. Writev as a linear combination of the unit vectors i

andj.


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PROPERTIES OF VECTORS IN SPACE

Let 321 vvv ,,v and 321 www ,,w be

vectors in 3 dimensional space and kis aconstant.

1. wv if and only if

332211wvwvwv ,, .

2. The magnitude ofv is2

3

2

2

2

1 vvv v

3. The unit vector in the direction ofv is

v

,,

v

v 321 vvv

4. 332211 wvwvwv ,,wv

5. 321 kvkvkvk ,,v

6. Zero vector is denoted as 0 000 ,, .


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Example:

Express the vector PQ if it starts at point

),,( 856P and stops at point ),,( 937Q incomponents form.

Example:

Given that 213 ,,a , 461 ,,b . Find

(a) ba 3 (b) b

(c) a unit vector which is in the direction ofb.

(d) find the unit vector which has the same

direction as ba 3 .


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Example: (PARALLEL VECTORS)

Vector w has initial point (2,-1,3) and terminal

point (-4,7,5). Which of the following vectors isparallel to w?

Example: (COLLINEAR POINTS)

Determine whether the point 0,1,2,3,2,1 QP and 6,7,4 R lie on the same line.


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THE DOT PRODUCT

(THE SCALAR PRODUCT)

The scalar product between two vectors

v = 321

,, vvv and w = 321

,, www is

cosvwv w

where is the angle between v and w.

Example:Ifv = 2i-j+k, w = i+j+2k and the angle

between v and w is 60, find wv .


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Theorem : (The Dot Product)

If 321 vvv ,,v and 321 www ,,w , then the

scalar product wv is

wv 321 vvv ,, 321 www ,,

332211 wvwvwv

Example:

Given that 3,2,2 u , 1,8,5v and

2,3,4 w , find

(a) vu (b) wvu (c) wvu

(d) the angle between u and v

(e) the angle between v and w.


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Example:

LetA=(4,1,2),B=(3,4,5) and C=(5,3,1) are the

vertices of a triangle. Find the angle at vertexA.


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Theorem :(Angle between two vectors)

The nature of an angle

, between two vectorsu and v.

1. is an acute angle if and only if 0 vu

2. is an obtuse angle if and only if 0 vu

3. = 90

if and only if 0 vu

Example:Show that the given vectors are perpendicular to

each other.

(a) i andj

(b) 3i-7j+2k and 10i+4j-k


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Theorem (Properties of Dot Product)

Ifu,v, and w are nonzero vectors and kis ascalar,1. uvvu

2. wuvuwvu

3. vuvu kk

4.2

vvv

5. 0 u00u


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THE CROSS PRODUCTS

(VECTOR PRODUCTS)

The cross product (vector product) vu is a

vector perpendicular to u and v whose

direction is determined by the right hand rule

and whose length is determined by the

lengths ofu and v and the angle between

them.

u

vvu


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Definition : (Cross Product)

Ifu and v are nonzero vectors, and ( 0 )

is the angle between u and v, then

sinvuvu ,

Theorem (Properties of Cross Product)

The cross product obeys the laws

(a) 0uu

(b) uvvu

(c) wuvuwvu

(d) vuvuvu kkk

(e) u //v if and only if 0u v

(f) u 0u00


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Theorem

If kiu 321 uuu and kiv 321 vvv ,

then

ji

ki

122113312332

321

321

()()(

vu

vuvuvuvuvuvu

vvv

uuu

Example:

1. Given that 4,0,3u and 2,5,1 v , find

(a) vu (b) uv

2. Find two unit vectors that are perpendicular

to the vectors u 2i+2j-3k and v = i+3j+k.


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THE LENGTH OF THE CROSS

PRODUCT AS AN AREA

B Cu

A D

v

Area of a parallelogram vuvu sin

Area of triangle = vu2

1

Example:

(a) Find an area of a parallelogram that is

formed from vectors u = i +j - 3k and

v = -6j + 5k.

(b) Find an area of a triangle that is formed

from vectors u = i +j - 3k and

v = -6j + 5k.


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SCALAR TRIPLE PRODUCT

Theorem

If 111 ,, zyxa , 222 ,, zyxb and 333 ,, zyxc ,

then

333

222

111

zyx

zyx

zyx

cba

Note: cbacba

Example:

If a = 3i + 4j - k, b = -6j+5k and c = i+j-k,

evaluate

(a) cba (b) cba

(c) bac (d) cab


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In this section we use vectors to study lines in

three-dimensional space.

.

HOW LINES CAN BE DEFINED USING

VECTORS?

The most convenient way to describe a line in

space is to give a point on it and a nonzero vector

parallel to it.

LINES IN SPACE


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Suppose Lis a straight line that passes through

),,( 000 zyxP and is parallel to the vector

kiv cba .

Thus, a point ),,( zyxQ also lies on the line if

vtQP

.


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Say OP0r and OQr

0rr PQ

vrr t 0

vrr t 0

cbatzyxzyx ,,,,,, 000

Theorem (Parametric Equations for a Line)

The line through the point ),,( 000 zyxP and

parallel to the nonzero vector cba ,,A has the

parametric equations

atxx 0 , btyy 0 , ctzz 0 , (1)


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If we let 000 ,, zyx0R denote the position

vector of ),,( 000 zyxP and zyx ,,R the

position vector of the arbitrary point Q(x,y,z) on

the line, then we write equation (1) in the vector

form

ARR t0

Example:

Give the parametric equations for the line

through the point (6,4,3) and parallel to the

vector 7,0,2 .


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Theorem (Symmetric Equations for a line)

The line through the point ),,( 000 zyxP and

parallel to the nonzero vector cba ,,A has

the symmetrical equations

c

zz

b

yy

a

xx 000

Example:

Given that the symmetrical equations of a line in

space is2

4

4

3

3

12

zyx, find

(a) a point on the line.

(b) a vector that is parallel to the line.


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ANGLE BETWEEN TWO LINES

Consider two straight lines

c

zz

b

yy

a

xxl 1111 :

and

f

zz

e

yy

d

xxl 2222 :

The line 1l parallel to the vector kiu cba

and the line2

l parallel to the vector

kiv fed . Since the lines 1l and 2l are

parallel to the vectors u and v respectively, then

the angle, between the two lines is given by

vu

vu cos


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Example:

Find an acute angle between line

1l = i + 2j + t(2i

j + 2k)

and line

2l = 2ij + k + s(3i6j + 2k).


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INTERSECTION OF TWO LINES

In three dimensional coordinates (space), two

line can be in one of the three cases as shown

below


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Example:


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SHORTEST DISTANCE

a) Distance From A Point To A Line

sinQPd

where is the angle between v and vector

QP

.

since

sinQPQP

vv ,

we have the shortest distance ofQ fromLas

v

v QPd


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Example:

Find the shortest path from the point Q(2, 0, -2)

to the line

2

1

1

1

3

2:

zyxl


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Suppose that is a plane. Point ),,( 000 zyxP

and ),,( zyxQ lie on it. If ki cbaN is a

non-null vector perpendicular (ortoghonal) to ,

thenNis perpendicular to PQ .

Thus, 0

NPQ 0,,,, 000 cbazzyyxx

0)()()( 000 zzcyybxxa

PLANES IN SPACES


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Conclusion:

The equation of a plane can be determined if a

point on the plane and a vector orthogonal to the

plane are known.

Theorem (Equation of a Plane)

The plane through the point ),,( 000 zyxP and

with the nonzero normal vector cba ,,N has

the equation

Point-normal form:

0000 zzcyybxxa

Standard form:

dcybyax with 000 czbyaxd


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Example:

1. Give an equation for the plane through the

point (2, 3, 4) and perpendicular to the vector

4,5,6 .

2. Give an equation for the plane through the

point (4, 5, 1) and parallel to the vectors

A= 1,0,2 and B 4,1,0 .

3. Give parametric equations for the line through

the point (5, -3, 2) and perpendicular to the

plane 5726 zyx .


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INTERSECTION OF TWO PLANES

Intersection of two planes is a line. (L)

To obtain the equation of the intersecting line,

we need

1) a point on the lineL which is given by

solving the equations of the two planes.

2) a vector parallel to the lineL which is

21 NN


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If cbaNN ,,21 , then the equation of the

lineL in symmetrical form is

c

zz

b

yy

a

xx 000

Example:


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ANGLE BETWEEN TWO PLANES

Properties of two planes

(a) An angle between the crossing planes is an

angle between their normal vectors.

21

21cos

NN

NN

(b) Two planes are parallel if and only if their

normal vectors are parallel, 21 NN

(c) Two planes are orthogonal if and only if

021 NN .

Example:

Find the angle between plane 043

yx andplane 522 zyx .


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ANGLE BETWEEN A LINE AND A

PLANE

Let be the angle between the normal vector N to a

plane and the lineL. Then we have

Nv

Nv

cos

where v is vector parallel toL. Furthermore, if the

angle between the lineL and the plane , then

2

2

cos2

sinsin


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Nv

Nv sin

Example:

Find the angle between the plane 523 zyx

and the line 3

3

1

2

2

3

zyx.


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SHORTEST DISTANCE FROM A

POINT TO A PLANE(a) From a Point To a Plane

Theorem

The distance D between a point ),,( 111 zyxQ

and the plane dczbyax is

222

111

N

N

cba

dczbyaxPQD

Where ),,( 000 zyxP is any point on the plane.

),,( 111 zyxQ

,,( 000 zyxP

D

N

D


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Example

Find the distance D between the point (1, -4, -3)

and the plane 1632 zyx

Example

(a) Show that the line

1

1

23

1

zyx

is parallel to the plane 123 zyx .

(b) Find the distance from the line to the plane

in part (a)


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(b) Between two parallel planes

The distance between two parallel planes

1dczbyax and 2dczbyax is given by

222

21

cba

ddD

Example

Find the distance between two parallel planes

322 zyx and 7442 zyx .


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(c) Between two skewed lines

Remark:

Both formula can also be used to compute the

distance between 2 skewed lines.


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Vectors

Example:

1. Find the shortest distance from P(1, -1, 2) to

the plane 3x7y +z = 5.

2. Find the shortest distance between the skewed

lines.

l1

:x = 1+2t,y = -1+ t,z = 2 + 4t

l2 :x = -2+4s,y = -3s ,z = -1+s

Chapter 3 Vectors[1]

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