CHAPTER 1 Polar Coordinates and Vector

8/3/2019 CHAPTER 1 Polar Coordinates and Vector

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POLAR COORDINATES &VECTORS

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EXAMPLE 1

Solution:

Form the Cartesian equation by eliminate parameter tfrom thefollowing equations

tx 2 14 2 ty

Given that , thus

Then

tx 2 2x

t

1

1

2

4

2

2

x

xy


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EXAMPLE 2

Solution:

Find the graph of the parametric equations

ttx 2 12 ty

We plug in some values oft.


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EXAMPLE 3

Find the graph of the parametric equations

tx cos ty sin 20 t


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TANGENT LINES TO PARAMETRIC CURVES

dtdx

dtdy

dx

dy

If 0and0 dt

dxdt

dyHorizontal

If0and0 dt

dxdt

dy Infinite slope

Vertical

If 0and0 dt

dxdt

dySingular points


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EXAMPLE 4

(a) Find the slope of the tangent line to the unit circle

at the point where

(b) In a disastrous first flight, an experimental paper airplane

follows the trajectory of the particle as

but crashes into a wall at time t= 10.

i) At what times was the airplane flying horizontally?

ii) At what times was it flying vertically?

tx cos ty sin

3

t

ttx sin3 ty cos34


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ARC LENGTH OF PARAMETRIC CURVES

b

a

dtdt

dy

dt

dxL

22

EXAMPLE 5

Find the exact arc length of the curve over the stated interval

2tx 3

3

1 ty 10 t)(a

tx 3cos ty 3sin t0)(b


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Consider the parametric equations,

a) Sketch the graph.

b) By eliminating t, find the Cartesian equation.

1239, for 3 2x t y t t

2R

EXAMPLE 6


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12

39, for 3 2x t y t t 29 9x y

)(a

Solution:

)(b


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Sketch the graph of 2 4 , 1 5 ,3So1ution:

In this form we can see that 2 4 , 1 5 , 3

Notice that this is nothing more than a 1ine, with

a point 2, 1,3 and a vector para11e1 is 4,5,1 .

F t t t t

x t y t z t

v

Graph in 3REXAMPLE 7


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I

sin

cos

tan

II

sin

tanIII

cosIV

THE TRIGONOMETRIC RATIOS


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For any angle ;

A

CT

S

+ve

-ve

tantan

coscos

sinsin



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Trigonometrical ratios of some special angles;

A1

2

O

B

30

60

3

B

A 1

1

O

45

45

2


1/ 21/ 2 3 / 2

3 / 2 1/ 2 1/ 2

1/ 3 3

30 45 60

sin 0 1

cos 1 0

tan 0 1 undefined

0 90


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Relationship between Polar and

Rectangular Coordinates

Ox

y P

sinry

cosrx

r

cosrx sinry

x

yyxr tan222


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Change the polar coordinates to Cartesian coordinates.

3

,2

3,1isscoordinateCartesianThe33sin2sin

13

cos2cos

then,3

and2Since

ry

rx

r

EXAMPLE 9

Solution:


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EXAMPLE 10

Find the rectangular coordinates of the point P whose

polar coordinates are

3

2,4,

r

Solution:

22

14

3

2cos4

x

322

343

2sin4

y

Thus, the rectangular coordinates ofP are 32,2, yx


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Change the coordinates Cartesian to polar coordinates. 1,1

4

7,2and

4

,2arescoordinatepolarpossibleThe

4

7or

4

,1tan

211

thenpositive,betochooseweIf

2222

x

y

yxr

r

1,1

x

y

EXAMPLE 11


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Symmetry Tests

SYMMETRIC CONDITIONS

about thex axis

about they axis

about the origin

,r ,r

,r

,r ,r

,r

,r ,r

,r


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),( r

),( r

),( r),( r

0

Symmetry with respect to x axis

Symmetry with respect to y axis


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),( r

),( r

Symmetry with respect to theorigin


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(c)Given that . Determine the symmetry of the

polar equation and then sketch the graph.sin33r

(d) Test and sketch the curve for symmetry.2sinr

(a) What curve represented by the polar equation 5r

(b) Given that . Determine the symmetry of the

polar equation and then sketch the graph.cos2r

EXAMPLE 13


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2cosr


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(a) Find the area enclosed by one loop of four petals 2cosr

(b) Find the area of the region that lies inside the circle

and outside the cardioid

sin3rsin1r

drAb

a

2

2

1

:8

Answer

:Answer

EXAMPLE 14


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Figure 10.4.1

(p. 730)


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Figure 10.4.14 (p. 735)


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Figure 10.4.22 (p. 738)


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Equations (12) (16) (p. 740)


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Equations (17) (19) (p. 740)


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x

y

z


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Graph the following ordered triples:

a. (10, 20, 10)b. (20, -10, -40)

EXAMPLE 15


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Distance Formula in3

R

The distance between and

is

21PP ),,( 1111 zyxP

),,( 2222 zyxP

2

12

2

12

2

1221 )()()( zzyyxxPP

Find the distance between (10, 20, 10) and

(-12, 6, 12).

EXAMPLE 16


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Vectors in3

R

A vector in R3is a directed line segment (an arrow)in space.

Given:-initial point

-terminal point

Then the vector PQ has the unique standardcomponent form

),,( 111 zyxP

),,( 222 zyxQ

121212 ,, zzyyxxPQ


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Standard Representation

of Vectors in the Space

The unit vector:points in the directions of the positivex-axispoints in the directions of the positivey-axispoints in the directions of the positivez-axis

i,j and k are called standard basis vector in R3.

Any vector PQ can be expressed as a linear combination ofi,j andk (standard representation ofPQ)

with magnitude

0,0,1i 0,1,0j 1,0,0k

kjiPQ)()()( 121212

zzyyxx

2

12

2

12

2

12 )()()( zzyyxx PQ


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Find the standard representation of the vector PQ

with initial point P(-1, 2, 2) and terminal point

Q(3, -2, 4).

EXAMPLE 17


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Parametric Form of a

Line in 3RIfL is a line that contains the point and

is parallel to the vector , thenL has

parametric form

Conversely, the set of all points that satisfysuch a set of equations is a line that passes

through the point and is parallel to a

vector with direction numbers .

),,( 000 zyx

kjiv cba

ctzzbtyyatxx 000

),,( zyx

),,( 000 zyx

],,[ cba


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Find parametric equations for the line that

contains the point and is parallel to thevector .

Find where this line passes through the

coordinate planes.

1, 1, 2 3 2 5 v i j k

EXAMPLE 18

Solution:

0 0 0

The direction numbers are 3, 2, 5 and

1, 1 and z 2, so the 1ine has the

parametric form

1 3 1 2 2 5

x y

x t y t z t


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25

2 11 9 11 9, ,

5 5 5 5 5

*This 1ine wi11 intersect the -p1ane when 0;

0 2 5 imp1ies

If , then and . This is the point 0 .


0 1 2 imp1ies

xy z

t t

t x y

xz y

t t

1

2

1 1 9 1 9

2 2 2 2 2

1

3

1 1 11 1 11

3 3 3 3 3

If , then and z . This is the point ,0, .


0 1 3 imp1ies

If , then and z . This is the point 0, , .

t x

yz x

t t

t y

continue solution:


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Symmetric Form of a Line in3

R

IfL is a line that contains the point and

is parallel to the vector

(A, B, and C are nonzero numbers), then the pointis onL if and only if its coordinates satisfy

kjiv cba

),,( 000 zyx

),,( zyx

czz

byy

axx 000


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Find symmetric equations for the lineLthrough the points

and .

Find the point of intersection with thexy-plane.

2,4, 3A 3, 1,1B

EXAMPLE 19


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0 0 0

The required 1ine passes through or and

is para11e1 to the vector

3 2, 1 4,1 3 1, 5,4 @ 5 4

Thus, the direction numbers are 1, 5,4 .

Let say we choose as , , .

2 4 3Then,

1 5 4

The sy

A B

A x y z

x y z

AB i j k

4 3mmetric equation is 2

5 4

y zx

Solution:


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11 1, ,4 4

This 1ine wi11 intersect the -p1ane when 0;3 4 3

2 and4 5 4

11 1

4 4The point of intersection of the 1ine with the -p1ane is 0 .

xy zy

x

x y

xy

continue solution:


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3R

1. Find the parametric and symmetric equations for the

point 1,0, 1 which is para11e1 to 3 4 .

2. Find the points of intersection of the 1ine

4 3 2 with each of the coordinate p1anes4 3

x yz

i j

.

3. Find two unit vectors para11e1 to the 1ine

1 2 52 4

x y z


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Line may Intersect, Parallel or Skew

Recall two lines in R2 must intersect if their slopes are

different (cannot be parallel)

However, two lines in R3 may have different direction

number and still not intersect. In this case, the lines are

said to be skew.

EXAMPLE 20


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In problems below, tell whether the two lines are

intersect, parallel, or skew . If they intersect, give thepoint of intersection.

3 3 , 1 4 , 4 7 ;

2 3 , 5 4 , 3 7

x t y t z t

x t y t z t

1

2 4 , 1 , 5 ;2

3 , 2 , 4 2

x t y t z t

x t y t z t

3 1 4 2 3 2;

2 1 1 3 1 1

x y z x y z

EXAMPLE 20

)(a

)(b

)(c

S l ti


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1 2

1

2

3 1 4 2 5 31. Let : and :

3 4 7 3 4 7

has direction numbers 3, 4, 7

and has direction numbers 3, 4, 7 .

Since both 1ines have same direction numbers

(or 3, 4, 7 = 3

x y z x y zL L

L

L

t

1 2

, 4, 7 , where 1),

therefore they are para11e1 or coincide.

Obvious1y, has point 3,1, 4 and has point 2,5,3 .4 7 , with the direction numbers 1,4,7 .

Because there is no ' ' for w

t

L A L B

a

AB i j k

hich 1,4,7 3, 4, 7 ,

the 1ines are not coincide, but just para11e1.

a

Solution:

S l i


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1

21 2

1

2

2 1 2 42. Let : and :

4 1 5 3 1 2

has direction numbers 4,1,5

and has direction numbers 3, 1, 2 .

Since there is no for which 4,1,5 3, 1, 2 ,

the 1ines are not pa

zx y x y zL L

L

L

t t

1

1 1 1 12

2 2 2 2

ra11e1 or coincide, maybe skew or intersect.

Express the 1ines in parametric form: 2 4 , 1 , 5 ;

: 3 , 2 , 4 2

L x t y t z t

L x t y t z t

Solution:

i l i


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1 2 1 2

1 2 1 2

1 7

1 2 1 22 2

1 2

Continue : 2At an intersection point we must have

2 4 3 4 3 2

1 2 3

5 4 2 5 2

So1ving the first two equations simu1taneous1y,

11 and 14 and since the so1ution is

t t t t

t t t t

t t t t

t t

not

satisfy the third equation, so the 1ines are skew.

continue solution:

S l i


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1 2

1

2

3 1 4 2 3 23. Let : and :2 1 1 3 1 1

has direction numbers 2, 1,1

and has direction numbers 3, 1,1 .

Since there is no for which 2, 1,1 3, 1,1 ,

the 1ines are not para

x y z x y zL L

L

L

t t

1 1 1 1

2 2 2 2

11e1 or coincide, maybe skew or intersect.

Express the 1ines in parametric form

: 3 2 , 1 , 4 ;

: 2 3 , 3 , 2

L x t y t z t

L x t y t z t

Solution:

i l i


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1 2

1 2 1 2

1 2

Continue : 3

At an intersection point we must have

3 2 2 3

1 3 1 and 1

4 2

Satisfy a11 of the equation,

then these two 1ines are intersect to each other.

The point of intersectio

t t

t t t t

t t

1

1 2 2 2

1

n is

3 2 3 2 1 1

1 1 1 2 or 2 3 , 3 , 2

4 4 1 3

1,2,3

x t

y t x t y t z t

z t

continue solution:

CLASS ACTIVITY 2


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CLASS ACTIVITY 2 :

In problems below, tell whether the two lines areintersect, parallel, or skew. If they intersect, givethe point of intersection.

1.

2.

3.

6 , 1 9 , 3 ;

1 2 , 4 3 ,

x t y t z t

x t y t z t

1 2 , 3 , 2 ;

1 , 4 , 1 3

x t y t z t

x t y t z t

1 2 3 2 1;

2 3 4 3 2

y z x y zx


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REMEMBER THAT

Theorem: The orthogonal vector theorem

Nonzero vectors v and n are orthogonal

(or perpendicular) if and only if

where n is called the normal vector.

0nv


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0 0 0

0 0

Let say, we have a p1ane containing point , , and

is orthogona1 (norma1) to the vector

So1ution:

If we have another any point , , in the p1ane, then

0

Q x y z

A B C

P x y z

Ai Bj Ck x x y y z

N i j k

N.QP

N.QP . i j

0

0 0 0

0 0 0

0 0 0 0 0 0

0 @

0, as ,

Then 0

z

A x x B y y C z z

A x x B y y C z z

Ax By Cz Ax By Cz D Ax By Cz

Ax By Cz D

k


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An equation for the plane with normal

that contains the point has the following forms:

Point-normal form:

Standard form:

Conversely, a normal vector to the planeis

A B C N i j k

0 0 0, ,x y z

0 0 0 0A x x B y y C z z 0Ax By Cz D

0Ax By Cz D A B C N i j k

EXAMPLE 21


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Find an equation for the plane that containsthe point P and has the normal vector Ngiven in:

1.

2.

1,3,5 ; 2 4 3P N i j k

1,1, 1 ; 2 3P N i j k

EXAMPLE 21


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Point-Normal form

Standard form

1,3,5 ; 2 4 3P N i j k

2 1 4 3 3 5 0x y z

2 1 4 3 3 5 02 2 4 12 3 15 0

2 4 3 5 0

x y z

x y z

x y z

1.

Solution :


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REMEMBER THAT..

Theorem: Orthogonality Property of The

Cross Product

Ifv and w are nonzero vectors in that are not

multiples of one another, then v x w isorthogonal to both v and w

3R

wvn

EXAMPLE 22


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Find the standard form equation of a

plane containing

and

1,2,1 , 0, 3,2 ,P Q

1,1, 4R


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0 0 0

Hint :

What we need?

?Point , , ?x y zN N PQ PR

Since, a11 point , and

are points in the p1ane,

so just pick one of them !!

P Q R

EXAMPLE 23


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0 0 0

0 0 0

Hint :

Equation for 1ine; , , ,

so, obvious1y, you just have to find

the va1ue of , and .

and , ,

x x At y y Bt z z Ct

A B C

x y z

Find an equation of the line that passes through the point

Q(2,-1,3) and is orthogonal to the plane 3x-7y+5z+55=0

N =Ai +Bj + Ck

(2, -1, 3)


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1. Find an equation for the p1ane that contains the

point 2,1, 1 and is orthogona1 to the 1ine

3 1.

3 5 2

2. Find a p1ane that passes through the point 1,2, 1

and is para11e1 to the p1ane 2 3 1.

3. Sh

x y z

x y z

1 1 2ow that the 1ine2 3 4

is para11e1 to the p1ane 2 6.

x y z

x y z

EXAMPLE 24


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Find the equation of a 1ine passing through 1,2,3

that is para11e1 to the 1ine of intersection of the p1anes

3 2 4 and 2 3 5.x y z x y z

Equation of a Line Parallel to The

Intersection of Two Given Planes

EXAMPLE 25


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Find the standard-form equation of the p1ane

determined by the intersecting 1ines.2 5 1 1 16

and3 2 4 2 1 5

x y z x y z

Equation of a Plane Containing

Two Intersecting Lines

EXAMPLE 26


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Find the point at which the 1ine with parametric

equations 2 3 , 4 , 5 intersects the

p1ane 4 5 2 18

x t y t z t

x y z

Point where a Line intersects with

a Plane.


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INTERSECTING PLANE

The acute anglebetween the planes :

21

21cosnn

nn

EXAMPLE 27

Find the acute angle of intersection between the

planes 4326and6442 zyxzyx

DISTANCE PROBLEMS INVOLVING


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DISTANCE PROBLEMS INVOLVING

PLANES

The distanceD between a point and theplane is

0000 ,, zyxP0 dczbyax

222

000

cba

dczbyaxD


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EXAMPLE 28


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Find the distanceD between the point (1,-4,-3) and the plane

1632 zyx


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(1) CircularCylinder

922 zx

three.radiusofcircle

aisgraphtheplane-On thexz


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(2) Ellipsoid

12

2

2

2

2

2

c

z

b

y

a

x


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12

2

2

2

2

2

c

z

b

y

a

x

(3) Hyperboloid of One Sheet


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(4) Hyperboloid of Two Sheets

12

2

2

2

2

2

c

z

b

y

a

x


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(5) Cone

02

2

2

2

2

2

c

z

b

y

a

x


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(6) Paraboloid

0c,2

2

2

2

czb

y

a

x

(7) H b li P b l id


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0c,2

2

2

2

czb

y

a

x

(7) Hyperbolic Paraboloid

EXAMPLE 29


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22

222

22

2

)1(z(v)1(iv)

16y(iii)

9z(ii)

1535(i)

yxzyx

x

y

zy

EXAMPLE 29

Sketch the graph of the following equations in 3-dimensions.

Identify each of the surface.


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Figure 11.8.3 (p. 833)


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Figure 11.8.4 (p. 833)


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Table 11.8.1 (p. 833)


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Table 11.8.2 (p. 835)

EXAMPLE 30


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(a) Convert from rectangular to cylindrical coordinates

(i) (-5,5,6) (ii) (0,2,0)

(b) Convert from cylindrical to rectangular coordinates

(c) Convert from spherical to rectangular coordinates

(d) Convert from spherical to rectangular coordinates

9,7(ii)3,6

,4)( ,i

4

,6

5(ii)2

,0,7)(

,i

3

2,

45(ii)0,0,3)(

,i


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Documents

CHAPTER 1 Polar Coordinates and Vector