MATH 37 Lecture Guide UNIT 5 (No Exercise)

7/29/2019 MATH 37 Lecture Guide UNIT 5 (No Exercise)

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MATH37LectureGuideUNIT5albabierra

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UNIT 5. VECTORS, LINES, PLANES and SURFACES MATH 37 LECTURE GUIDE

Objectives: By the end of the unit, a student must correctly and confidently be able to:

enumerate and apply properties of vectors in the plane and in space;

perform and interpret vector operations;

find the equations of a line and equation of a plane in space; and

identify and sketch cylinders and quadric surfaces.__________________________

5.1 Vectors in 2D and in 3D.

Vectors are quantities that represent both magnitude and direction.

Geometric representation: directed line segments (or arrows) where

magnitude = length of the arrow direction of the vector = direction of the arrow

Directions are measured by the angle a segment makes with the horizontal. Positive if measuredcounter-clockwise. Negative if measured clockwise.

initial point= arrow tail terminal point= arrow head

A vector can have several representations on a plane (depending on the initial and terminalpoints).A vector is in its position representation if its initial point is at the origin. Its direction is

identified by the angle it makes with the positive axis in the counter-clockwise direction.

TO DO!!! Consider a vector with initial point

at ( )42 , and terminal point at ( )65, .Determine the magnitude and direction of thisvector.

TO DO!!! The following are differentrepresentations of the same vector.

Initial point Terminal point

A ( )23, ( )11 ,

B ( )15 , ( )41 ,

C ( )51, ( )25,

Position representation:


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MUST REMEMBER!!! In a humane way, if as +n , La n , then.

Given vector b,a . In its position representation, the terminal point is at ( )b,a .

Equality of vectors.

Two vectors b,a and d,c are equal if and only if ca = and db = .

Two vectors are equal if their magnitudes and directions are equal.

The direction angle A is consistent with the angle used in polar coordinates.

Definitions.

A vector in the plane is an ordered pair of real numbers b,a . The numbers a and b are

called the components of the vector.

TO DO!!! Determine the components of the following vectors.

1. Vector with initial point at ( )42 , and terminal point at ( )65, .

2. Vector with initial point at ( )23 , and terminal point at ( )11 , .

MUST REMEMBER!!! If the initial point of a vector is at ( )11 y,x and its terminal point is at

( )22 y,x , then its components are given by ________________.

Definitions.

The magnitude of a vector , denoted by A , is the length of any of its representation. The

direction angle of a nonzero vector , denoted by A , is the measure of the angle formed by

the vector (in position representation) with the positive axis.

REMEMBER!!! If b,aA = , then 22 baA += andx

ytan A = .

Also, AA sinA,cosAA = .


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Now, vectors in 3D!

TO DO!!!

1. Determine the magnitude and direction angle of the vectors 44 ,A and 31 ,B .

2. Determine the components of the vector with a magnitude of6 units in the direction of

3

5 = .

Definition.

The set of all ordered triples of real numbers is called as the three-dimensional number

space, denoted by 3R . Each ordered triple ( )z,y,x is called as a point in the three-dimensional space.

Definition. A unit vectoris a vector with a magnitude of 1 unit.

REMEMBER!!!Unit vectors in the direction of the positive and y axes, respectively:

01,i = and 10,j =

Given b,aA = . bjaiA +=

unit vector in the direction of , denoted by AU :

A

b,

A

aUA = or AAA sin,cosU = .

TO DO!!!

1. Determine the unit vector in the direction of 512 , .

2. Determine the unit vector in the direction of the vector with a magnitude of 10 units in the

direction of6

= .


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In its position representation, vector c,b,a has its initial point at the origin while the terminal point

is at ( )c,b,a .

z

x

y

REMEMBER!!! Given points ( )1111 z,y,xP and ( )2222 z,y,xP .

The distance between these points (or the length of line segment 21PP ) is given by

( ) ( ) ( )2

122

122

1221 zzyyxxPP ++= .

The midpoint of the line segment 21PP is given by

+++

222

212121

21

zz,

yy,

xxM PP

Definition.

A vector in three-dimensional space is an ordered triple of real numbers c,b,a . The

numbers b,a and c are called the components of the vector.

TO DO!!!

Determine the distance between the points

( )4231 ,,P and ( )2342 ,,P . Also,

determine the midpoint of segment21PP .


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The direction angles

Definitions.

The magnitude of a vector , denoted by

A , is the length of any of its

representation.

The direction anglesof a nonzero vectorare the three angles that have the

smallest nonnegative radian measure ,

and measured from the positive side of

the , y and z axes, respectively, tothe position representation of the vector.

REMEMBER!!!

If vector has its initial point at ( )111 z,y,x and terminal point at ( )222 z,y,x ,

then 121212 zz,yy,xxA = .

If c,b,aA = , then 222 cbaA ++= .

If , and are the direction angles, thenA

acos = ,

A

bcos = and

A

ccos = .

where 1222 =++ coscoscos .

z

x

y

REMEMBER!!!Unit vectors in the direction of the positive , y and z axes, respectively:

001 ,,i = , 010 ,,j = and 100 ,,k =

Given c,b,aA = . ckbjaiA ++=

unit vector in the direction of , denoted by AU :

A

c,

A

b,

A

aUA =

TO DO!!! Consider vector with initial point at ( )314 ,, and terminal point at ( )542 ,, .Determine the components of this vector. Also, determine its magnitude and the cosine of thedirectional angles. Also, determine the unit vector in the direction of .


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5.2 Operations on Vectors.

Geometrically, the sum and difference of two vectors can be represented by the parallelogrammethod. The negative of a vector is the vector of the same magnitude but towards the opposite

direction. Scalar multiplication "stretches" ( if 1>c )or "shrinks" ( if 1


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TWO MORE OPERATIONS . . .

MUST!!! Geometrically, interpret a dot product!

Operations on vectors. Consider vectors 321 a,a,aA= and 321 b,b,bB= .

Sum of two vectors: 332211 ba,ba,baBA +++=+

Negative of a vector: 321 a,a,aA =

Difference of two vectors: 332211 ba,ba,baBA =

Scalar multiplication: If c is a scalar, then 321 ca,ca,cacA=

TO DO!!! Evaluate the following.

1. 4232 ,, 2. 423321 ,,,, .

REMEMBER!!! If and are nonzero vectors, then ABcosBABA = where AB is

the (smallest nonnegative) angle (in radian measure) between the two vectors.

If two vectors and are in the same direction, 0=AB .

If two vectors and are in opposite directions, =AB .

Definition.

The dot product of vectors 21 a,aA= and 21 b,bB= is given by 2211 babaBA += .

In the case of vectors in three-dimensional space, given 321 a,a,aA= and

321 b,b,bB= , 332211 bababaBA ++= .

TO DO!!! Consider 322 ,,A = and 024 ,,B = .

Determine the unit vector in the direction of BA 23 .


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On scalar and vector projections . . .

If and are nonzero vector, the scalar projection of onto is given byA

BA ,

and the scalar projection of onto is given by B

BA .

Also, AB is the angle between vectors and ,

the scalar projection of onto is given by ABcosB , and

the scalar projection of onto is given by ABcosA .

Remarks: A scalar projection is positive if the projection is in the same direction. A scalar projection is negative if the projection is in the opposite direction.

Scalar projection of onto Vector projection of onto

TO DO!!! Determine the angle between the following pairs of vectors.

1. 54 , and 125 , 2. 112 ,, and 531 ,,

(Supplement) Some properties

If , and C are any vectors in the plane and c is any scalar, then

i. = (commutativity)

ii. ( ) CABACBA +=+ (distributivity)iii. ( ) ( ) BcABAc =

iv. = , where 00,=

REMEMBER!!! Two non-zero vectors are orthogonal (orperpendicularwith each other) if andonly if their dot product is 0 (zero).


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The vector projection of onto is the vector with a magnitude of the scalar projection in thedirection of .

The vector projection of onto is given by

A

A

A

BA.

The vector projection of onto is given by

B

B

B

BA.

Projection is an important tool in engineering like in doing isometric views of structures.

Definition. (Cross product)

If 321 a,a,aA= and 321 b,b,bB= , then the cross product of and

is given by

321

321

bbb

aaa

kji

BA = kbb

aaj

bb

aai

bb

aa

21

21

31

31

32

32 +=

TO DO!!!

1. If 212 ,,A = and 033 ,,B = , evaluate .

TO DO!!! Determine the angle between the following pairs of vectors.

1. Consider vectors 24,A and 43 ,B . Evaluate the following.

a. vector projection of onto

b. vector projection of onto

2. Determine the scalar and vector projections of 036 ,,A = onto 424 ,,B = .


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__________________________

5.3 Planes and Lines in

3

R (TC7 pp. 861-871 / TCWAG pp. 861-872)

IfN is a given non-zero vector and 0P is a point,

then the set of all points for whichPP0 and N

are orthogonal is defined to be a plane through

0P and having N as a normal vector.

Exercise.

Consider ( )032 ,,P , ( )150 ,,Q and ( )301 ,,R .

Determine a vector that is orthogonal to vectorsPQ and

QR .

(Supplement) Some properties

1. If is any vector in the three-dimensional space, then

= ==

2. = ii = jj =kk

kji = ikj = jik =

kij = ijk = jki =

3. ( )ABBA =

4. If , and C are vectors identifying nonparallel sides of some parallelepiped,

then ( ) CBA is the volume of the parallelepiped.

PLANE

2. If 402 ,,M = and 120 ,,N= , evaluate N .

REMEMBER!!! Geometrically, if and are nonzero vectors, then

is a vector orthogonal (or perpendicular) to both and .


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Two planes are parallel if and only if their normal vectors are parallel.

Two planes are perpendicular if and only if their normal vectors are orthogonal.

IfR is a given non-zero vector and 0P is a point,

then the set of all points for whichPP0 is parallel

to R is a line through 0P and parallel to R .

TO DO!!! Determine the equation of the following planes.

1. plane through the point ( )324 ,, and perpendicular to the vector 312 ,,

2. plane through the point ( )471 ,, and parallel to the plane 0532 =+ zyx

3. plane containing the points ( )032 ,,P , ( )150 ,,Q and ( )301 ,,R

REMEMBER!!!

1. If ( )0000 z,y,xP is a point in a plane having c,b,a as a normal vector, then thestandard equation of the plane is given by

( ) ( ) ( ) 0000 =++ zzcyybxxa .

2. If b,a and c are not all zero, the graph of the equation 0=+++ dczbyax is a plane

having c,b,a as a normal vector.


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__________________________

5.4 Cylinders and Quadric Surfaces (TC7 pp. 886-900 / TCWAG pp. 883-894)

REMEMBER!!! The standard equation of a sphere of radius r and centered at ( )l,k,h is

given by ( ) ( ) ( ) 2222 rlzkyhx =++ .

Remark: The graph in 3R defined by 0222 =++++++ DzCByAxzyx is either a sphere, a

point or the empty set.

Let L be a line that contains the point ( )0000 z,y,xP and is parallel to the vector

c,b,aR = . Using t as a parameter, the parametric equations ofL is given by

atxx += 0 btyy += 0 ctzz += 0

If none of b,a and c is zero, the symmetric equations ofL is given by

c

zz

b

yy

a

xx 000 =

=

TO DO!!! Determine the parametric and symmetric, if possible, equations of the following lines.

1. line L passing through ( )321 ,, and which is parallel to the vector 542 ,, .

2. line containing the points ( )032 ,, and ( )154 ,, .

TO DO!!! Identify the graphs of the following equations.

1. 03422222 =+++++ zyxzyx

2. 0542222 =++++ zxzyx

3. 0742222 =++++ zyzyx


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Cylinder defined by 2522 =+ yx

The general equations for a quadric surface can be written in standards form for a more efficientidentification of the graphs.

REMEMBER!!!

In the three-dimensional space, the graph of

an equations in two of the three variables ,y and z is a cylinder.

TO DO!!!Sketch the following cylinders.

1. ysinz = 2. 42 = xz

z

x

y

z

x

y

z

x

y

Definition.

The graph of a second-degree equation in three variables , y and z

0222 =++++++++++ JHIzHyGxFyzExzDxyCzByAx

is called a quadric surface.

REMEMBER!!!

To sketch graphs of quadric surfaces, obtain the traces of the surface on the y plane ( 0=z ),

yz plane ( 0=x ) and z plane ( 0=y ). Level curves at particular values of z can also beused.


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SOME standard forms

Ellipsoid: 012

2

2

2

2

2

>=++ c,b,ac

z

b

y

a

x

Elliptic hyperboloid of one sheet: 01

2

2

2

2

2

2

>=+ c,b,ac

z

b

y

a

x

Elliptic hyperboloid of two sheets: 012

2

2

2

2

2

>=+ c,b,ac

z

b

y

a

x

Elliptic cone: 002

2

2

2

2

2

>=+ c,b,ac

z

b

y

a

x

Elliptic paraboloid: 0002

2

2

2

>=+ c,b,ac

z

b

y

a

x

Hyperbolic paraboloid: 0002

2

2

2>= c,b,a

c

z

b

y

a

x

0012

2

2

2

>= c,b,ac

z

b

y

a

x


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TO DO!!! Sketch the following quadric surfaces by using the traces over the three major planes.

1. 11644

222

=++zyx

Traces:

y plane yz plane z plane

2. 194

222 =

zyx

Traces:



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3. 19

222 =++

zyx

Traces:


4. 122 =+ zyx (What if 022 =+ zyx ?)

Traces:


_____________________________ END OF UNIT 5 Lecture Guide

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MATH 37 Lecture Guide UNIT 5 (No Exercise)