POLAR COORDINATES &

VECTORS

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Suppose that a particle moves along a curve C in the

xy-plane in such a way that its x and y coordinates, as

functions of times are

The variable t is called the parameter for the

equations.

)(tfx )(tgy

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EXAMPLE 1

Solution:

Form the Cartesian equation by eliminate parameter t from the

following equations

tx 2 14 2 ty

Given that , thus

Then

tx 22

xt

1

12

4

2

2

x

xy

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EXAMPLE 2

Solution:

Find the graph of the parametric equations

ttx 212 ty

We plug in some values of t .

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EXAMPLE 13

Find the graph of the parametric equations

tx cos ty sin 20 t

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x

y

z

Graph the following ordered triples:

a. (3, 4, 5)

b. (2, -5, -7)

EXAMPLE 1

Distance Formula in3R

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 2

Vectors in3R

A vector in R3 is a directed line segment (“an arrow”) in space.

Given:

-initial point

-terminal point

Then the vector PQ has the unique standard component form

),,( 111 zyxP

),,( 222 zyxQ

121212 ,, zzyyxxPQ

Standard Representation

of Vectors in the Space

The unit vector:points in the directions of the positive x-axispoints in the directions of the positive y-axispoints in the directions of the positive z-axis

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

Any vector PQ can be expressed as a linear combination of i, j and k (standard representation of PQ)

with magnitude

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

1,0,0k

kjiPQ )()()( 121212 zzyyxx

2

12

2

12

2

12 )()()( zzyyxx PQ

Find the standard representation of the vector PQ

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

Q(3, -2, 4).

EXAMPLE 3

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(1) Circular

Cylinder922 zx

three.radius of circle

a isgraph theplane- On thexz

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(2) Ellipsoid

12

2

2

2

2

2

c

z

b

y

a

x

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(3) Paraboloid

0c ,2

2

2

2

czb

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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1 2

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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0c ,2

2

2

2

czb

y

a

x

(7) Hyperbolic Paraboloid

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22

222

22

2

)1(z (v)

1 (iv)

16y (iii)

9z (ii)

1535 (i)

yx

zyx

x

y

zy

EXAMPLE 29

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

Identify each of the surface.

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Parametric Form of a

Line in3R

If L is a line that contains the point and

is parallel to the vector , then L has

parametric form

Conversely, the set of all points that satisfy

such a set of equations is a line that passes

through the point and is parallel to a

vector with direction numbers .

),,( 000 zyxkjiv cba

ctzzbtyyatxx 000

),,( zyx

),,( 000 zyx

],,[ cba

Find parametric equations for the line that

contains the point and is parallel to the

vector .

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

2

5

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 .

*This 1ine wi11 intersect the -p1ane when 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, .

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

0 1 3 imp1ies

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

t x

yz x

t t

t y

…continue solution:

Symmetric Form of a Line in3R

If L is a line that contains the point and

is parallel to the vector

(A, B, and C are nonzero numbers), then the point

is on L if and only if its coordinates satisfy

kjiv cba

),,( 000 zyx

),,( zyx

c

zz

b

yy

a

xx 000

Find symmetric equations for the line Lthrough the points

and .

Find the point of intersection with the

xy-plane.

2,4, 3A

3, 1,1B

EXAMPLE 19

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:

11 1, ,

4 4

This 1ine wi11 intersect the -p1ane when 0;

3 4 32 and

4 5 4

11 1

4 4

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

xy z

yx

x y

xy

…continue solution:

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 p1anes

4 3

x yz

i j

.

3. Find two unit vectors para11e1 to the 1ine

1 25

2 4

x yz

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.

In problems below, tell whether the two lines are intersect, parallel, or skew . If they intersect, give the point 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

12 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

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:

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:

1 2 1 2

1 2 1 2

1 7

1 2 1 22 2

1 2

Continue : 2

At 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:

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:

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 :

In problems below, tell whether the two lines are intersect, parallel, or skew. If they intersect, give the 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

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

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

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 plane

is

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

Find an equation for the plane that contains the 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

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 0

2 2 4 12 3 15 0

2 4 3 5 0

x y z

x y z

x y z

1.

Solution :

REMEMBER THAT..

Theorem: Orthogonality Property of The Cross Product

If v and w are nonzero vectors in that are not multiples of one another, then v x w is orthogonal to both v and w

3R

wvn

Find the standard form equation of a plane containing and

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

1,1, 4R

EXAMPLE 22

0 0 0

Hint :

What we need?

?

Point , , ?x y z

N N PQ PR

Since, a11 point , and

are points in the p1ane,

so just pick one of them !!

P Q R

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

EXAMPLE 23

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)

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 1ine

2 3 4

is para11e1 to the p1ane 2 6.

x y z

x y z

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 24

Find the standard-form equation of the p1ane

determined by the intersecting 1ines.

2 5 1 1 16 and

3 2 4 2 1 5

x y z x y z

Equation of a Plane Containing Two Intersecting Lines

EXAMPLE 25

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.

EXAMPLE 26

INTERSECTING PLANE

The acute angle

between the planes :

21

21cos

nn

nn

EXAMPLE 27

Find the acute angle of intersection between the

planes 4326 and 6442 zyxzyx

DISTANCE PROBLEMS INVOLVING

PLANES

The distance D between a point and the

plane is

0000 ,, zyxP0 dczbyax

222

000

cba

dczbyaxD

EXAMPLE 28

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

1632 zyx

A polar coordinate system consists of :

A fix point O, called the pole or origin

Polar coordinates where

r : distance from P to the origin

: angle from the polar axis to the ray OP

),( r

),( rP

Polar axisOrigin

O

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I

sin

cos

tan

II

sin

tan

III

cos

IV

THE TRIGONOMETRIC RATIOS

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For any angle θ;

A

CT

S

θ

θ+ve

-ve

tantan

coscos

sinsin

THE TRIGONOMETRIC RATIOS

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Trigonometrical ratios of some special angles;

A1

2

O

B

30°

60°

3

B

A1

1

O

45°

45°

2

THE TRIGONOMETRIC RATIOS

1/ 2 1/ 2 3 / 2

3 / 2 1/ 2 1/ 2

1/ 3 3

θ 30° 45° 60°

sin θ 0 1

cos θ 1 0

tan θ 0 1 undefined

0 90

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EXAMPLE 4

Plot the points with the following polar coordinates

0225 ,3 )(a

3 ,2 )(

bSolution:

(a)0225

0225,3P

xO

)(b

3,2

P

xO

3

Relationship between Polar and

Rectangular Coordinates

Ox

y P

sinry

cosrx

r

cosrx sinry

x

yyxr tan 222

Change the polar coordinates to Cartesian coordinates.

3,2

3,1 is scoordinateCartesian The

33

sin2sin

13

cos2cos

then,3

and 2 Since

ry

rx

r

EXAMPLE 5

Solution:

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EXAMPLE 6

Find the rectangular coordinates of the point P whose

polar coordinates are

3

2,4,

r

Solution:

22

14

3

2cos4

x

322

34

3

2sin4

y

Thus, the rectangular coordinates of P are 32,2, yx

Change the coordinates Cartesian to polar coordinates. 1,1

4

7,2 and

4,2 are scoordinatepolar possible The

4

7or

4

,1tan

211

thenpositive, be to choose weIf

2222

x

y

yxr

r

1,1

x

y

EXAMPLE 7

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EXAMPLE 8

Find polar coordinates of the point P whose rectangular

coordinates are 4,3

Solution:

5or 5 Thus

2543 22222

rr

yxr

Then,

3

4

3

4tan

x

y

Therefore, 000 13.23313.53180

Symmetry Tests

SYMMETRIC CONDITIONS

about the x axis

about the y axis

about the origin

,r ,r

,r

,r ,r

,r

,r ,r

,r

),( r

),( r

),( r

),( r

0

Symmetry with respect to x axis

Symmetry with respect to y axis

),( r

),( r

Symmetry with respect to the origin

(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 9

2cosr

(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

drA

b

a

2

2

1

:8

Answer

:Answer

EXAMPLE 10

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TANGENT LINES TO PARAMETRIC CURVES

dtdx

dtdy

dx

dy

If 0 and 0 dt

dxdt

dyHorizontal

If 0 and 0 dt

dxdt

dy Infinite slope

Vertical

If 0 and 0 dt

dxdt

dySingular points

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EXAMPLE 14

(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

ARC LENGTH OF PARAMETRIC CURVES

b

a

dtdt

dy

dt

dxL

22

EXAMPLE 15

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

2tx 3

3

1ty 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.

12

39, for 3 2x t y t t

2REXAMPLE 16

12

39, for 3 2x t y t t

29 9x y

)(a

Solution:

)(b

Sketch the graph of 2 4 , 1 5 ,3

So1ution:

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 17

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sCoordinatePolar lCylindrica

cosrx

sinry

22 yxr

x

y1tan

r0

20

z

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sCoordinatePolar Spherical

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Figure 11.8.3 (p. 833)

Figure 11.8.4 (p. 833)

Table 11.8.1 (p. 833)

Table 11.8.2 (p. 835)

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EXAMPLE 30

(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 cylindrical coordinates

9,7 (ii) 3,6

,4)( ,πi

4,

65 (ii)

2,0,7)(

π,i

3

2,

45 (ii) 0,0,3)(

π,i

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