Vectors in 2D and 3D Vectors 1. Three dimensional ...math.bu.edu/people/mkon/ma242/L2.pdf ·...

Vectors in 2D and 3DVectors1. Three dimensional coordinates:

Point in plane: two perpendicular lines asreference:

Vectors in 2D and 3D

Point in space: three perpendicular lines asreference -


B C B C plane plus z axis perpendicular to plane.

Coordinates of point indicated aboveT À ÐBß Cß DÑ

[ ]e.g., three corner lines of the room

[we get the three numbers by droppingÐBß Cß DÑperpendiculars to the three axes]

Vectors in 2D and 3DEx 1: Calculation of an orbit to the moon:

[need to set up coordinate system and measure allpoints elative to this r ]


[all mathematics is done by specifying position ofthe spacecraft and the moon relative to somecoordinate system, say centered at the earth; that'show orbits are calculated at JPL and NASA]

Vectors in 2D and 3D2. Vectors:

A vector is an arrow - it has direction and length. Ifyou are hiking and say that you are 3 mi NNW ofyour camp you are specifying a vector.

Vectors in 2D and 3DThe precise mathematical statement is that:

Geometric definition of vectors: A is avectordirected line segment. The length of a vector isvsometimes called its or the of .magnitude norm vWe will always abbreviate length by the symbol

length of v vœ l l Þ

Vectors in 2D and 3DTwo vectors are equal if they point in the samedirection and have the same length: [where the vector starts is not important]

Vectors in 2D and 3DWe can add vectors:

Vectors in 2D and 3DAnd we can multiply vectors by real numbers(scalar multiplication):

If then is the vector in theα α !ß adirection of whose length is .a aα

If then satisfiesα α ! a

direction direction Ð Ñ œ Ð ÑÞαa a

l l œ l ll l Þα αa a

Vectors in 2D and 3DSince we can add vectors and perform scalarmultiplication, we can subtract two vectors:

-a b a b œ Now assume initial points of all vectors are locatedat the origin:

Vectors in 2D and 3DConsider also the unit vectors e e e" # $ß ß À


[We see and ]+ + + + +" " # # " " # # $ $e e e e e

Notation: a e e eœ + + + œ œ+++

" " # # $ $

"

#

$

Ô ×Õ Ø

vector from origin to Ð+ ß + ß + ÑÞ" # $

Given the vector we geta e e e œ + + + ß" " # # $ $

by distance formula:

l l œ Ð+ !Ñ Ð+ !Ñ Ð+ !Ñ œa È " # $# # #È+ + +" # $

# # #


Analytic definition of vectors in 3 dimensions: A vector

is an array of numbers.Ô ×Õ Ø+++

"

#

$

Vectors in 2D and 3D3. Addition of vectors

Analytic addition of vectors:

a bœ œ+ ,+ ,” • ” •" "

# #

fig 4


a b œ+ ,+ ,” •" "

# #

Simliarly in three dimensions:

fig 5


a bœ œ+ ,+ ,+ ,

Ô × Ô ×Õ Ø Õ Ø

" "

# #

$ $

Ê œ œ+ ,+ ,+ ,

a bÔ ×Õ Ø

" "

# #

$ $

Moreover, can show similarly that:

αααα

a œ+++

Ô ×Õ Ø

"

#

$

Vectors in 2D and 3DExample 2:

A light plane flies at a heading of due north(direction which airplane is pointed) at air speed(speed relative to the air) of 120 km/hr in a windblowing due east at 50 km/hr. What direction andspeed does the plane move at relative to theground?

A: Define the velocity of the airplane as the vectorv whose length is the speed of the plane andwhose direction is the direction of the plane:


Note: wind velocity is ; plane's air velocity500” •

(relative to wind) is 0120” •

Also: every hour plane flies 120 kilometers northand 50 kilometers east.

Thus direction of the plane is same as 50120” •


fig 7

What is the speed: È"#! &! œ "$!Þ# #


Thus speed of airplane is 130 = length of ;” •&!"#!

direction of airplane is same as 50120” •

Ê velocity of airplane is vector

= + = wind velocity + air50 50 0120 0 120” • ” • ” •

velocity of plane

Vectors in 2D and 3DMoral: when we take an object moving at velocityv" (air velocity) in a medium which is moving atvelocity (wind velocity), the total velocity of thev#

object is .v v vœ " #

Properties of Vectors4. Properties of vectors:Theorem: If , and are vectors (in two or threea b c,dimensions) and and are scalars, then:. /(a) (commutativity)a b b a œ (b) (associativity)Ð Ñ œ Ð Ña b c a b c(c) (distributivity).Ð œ . .a b a bÑ(d) (distributivity)Ð. /Ñ œ . /a a a(e) There is a unique vector such that for0 a 0 a œall vectors a(f) For every vector there is a unique vector a a such that a a œ !(g) .Ð/ Ñ œ Ð./Ña a(h) " † œa a

Properties of Vectors

[other properties you would expect are listed in the book;they follow from the above properties]

Proof: (a): Assume that

a bœ à œ+ ,ã ã+ ,


" "

8 8

Note that the entries in have the general form for a + 3 œ "3

through .8

Properties of VectorsWe will use shorthand by writing in terms of the generalaterm in :a

a œ Ð+ ÑÞ3

Similarly,

b œ Ð, ÑÞ3

Thus

a b b a œ œ œ œ Þ+ , + , , +ã ã ã ã+ , + , , +

Ô × Ô × Ô × Ô ×Õ Ø Õ Ø Õ Ø Õ Ø

" " " " " "

8 8 8 8 8 8

Properties of VectorsIn shorthand the above could be written:

a b b a œ Ð+ Ñ Ð, Ñ œ Ð+ , Ñ œ Ð, + Ñ œ Ð, Ñ Ð+ Ñ œ Þ3 3 3 3 3 3 3 3

(b)

Ð Ñ œ Ð Ð+ Ñ Ð, ÑÑ Ð- Ñ œ Ð+ , Ñ Ð- Ña b c 3 3 3 3 3 3

œ ÐÐ+ , Ñ - Ñ œ Ð+ Ð, - ÑÑ œ Ð+ Ñ Ð, - Ñ3 3 3 3 3 3 3 3 3

œ + ÐÐ, Ñ Ð- ÑÑ œ Ð Ñ3 3 3 a b c as desired.

Properties of VectorsAs an exercise, write out the proof in full, i.e. replace

Ð+ Ñ Ä

++ã+

3

"

#

8

Ô ×Ö ÙÖ ÙÕ Ø

Ð+ , Ñ Ä ß

+ ,+ ,

ã+ ,

3 3

" "

# #

8 8


etc., to see how it looks.

Properties of Vectors

[Note it is remarkable that if you replace the word vectorwith the word matrix, the same statements as above are allstill true.]

Vectors in n dimensionsAnalytic definition of vectors in dimensions: 8 A vectoris a vertical array of numbers:8

v œ Þ

++ã+


"

#

8

Note: all definitions of analytic operations on vectors in 3dimensions hold for vectors in dimensions.8

Vectors in n dimensionsCan easily see that all properties (1)-(8) of vectors in 3 and2 dimensions carry over to vectors in dimensions; proofs8are identical.

[geometric visualization of vectors in dimensions is not8necessary at this point]

Vector spans5. Spans of vectors

Def 6: We define ‘ ‘$"

#

$

3œ B −BBB

Ú ÞÛ ßÜ à

Ô ×Õ Ø»

where means the set of all real numbers. Thus is all‘ ‘$

3-tuples of real numbers.

Def 7: A of two vectors and is a sum linear combination a b- -" #a b

for constants and - - Þ" #

Linear combination for larger collection of vectorsworks the same way.

Vector spans

Ex: Consider two vectors a e b eœ œ à œ œ À" !! "! !

Ô × Ô ×Õ Ø Õ Ø" #

Vector spansThen:

- - œ - - œ Þ" ! -! " -! ! !

" # " #

"

#a bÔ × Ô × Ô ×Õ Ø Õ Ø Õ Ø

Note as and vary, this covers points in the - - B C" # allplane.

We say of and all possible linear combinationsspan a b œof and all vectors in plane.a b œ B C

Vector spansGenerally can show: span of two vectors all vectorsœcontained in the plane of the first two.

General definition: The of a collection of vectorsspan Ö ß ßá ß ×v v v" # 8 is the collection of all possible linearcombinations

spanÖ ßá ß × œ Ö- - á - À - − ×Þv v v v v v"ß # 8 " " # # 8 8 3 ‘

Above means "is an element of"− Þ

Matrices and linear systems6. Matrices and linear systems of equations:

Return to previous problem: Coal, Steel,Electricity:

[NOTE: There is an example of Leontief'seconomic model in the book; the equations thereare DIFFERENT than here; they guarantee that theamount used for the production of each commodity(coal, steel, electricity) ends up equalling the valueof the commodity produced; in our variation ofLeontief's equations we have (non-zero) targets fornet production of each commodity, and findproduction levels to attain those targets]

Matrices and linear systems

To make $ of coal, takes no coal, $. of steel, " "! $. of electricity."!To make $ of steel, it takes $. of coal, $. of" #! "! steel, and $. of electricity.#!To make $ of electricity, it takes $. of coal; $. " %! #! of steel, and $. of electricity."!

Let total amount of coal producedB œ"

total amount of steel producedB œ#

total amount of electricity producedB œ$

Matrices and linear systemsGoals of economy:

If we want the economy to output $1 billion of coal,$.7 billion of steel, and $2.9 billion of electricity,how much coal, steel, and electricity will we need touse up? I.E., what will be?B ß B ß B" # $

Recall the equations:B Þ#B Þ%B œ "" # $

(*) Þ"B Þ*B Þ#B œ Þ(" # $

Þ"B Þ#B Þ*B œ #Þ*" # $

Matrices and linear systems give us for and B 3 œ "ß #ß $Þ3

Note: system (*) equivalent to a single matrixequation:

Ô × Ô × Ô ×Õ Ø Õ Ø Õ Ø

" Þ# Þ% B " Þ" Þ* Þ# B Þ( Þ" Þ# Þ* B #Þ*

œ"

#

$

Matrices and linear systemsCheck:Ä


B Þ#B Þ%B " Þ"B Þ*B Þ#B Þ( Þ"B Þ#B Þ*B #Þ*

œ" # $

" # $

" # $

Ê B Þ#B Þ%B œ " " # $

(*) Þ"B Þ*B Þ#B œ Þ( ß" # $

Þ"B Þ#B Þ*B œ #Þ*" # $

Matrices and linear systems = original system of equations.

[Very compact form for writing equations.]

If we call

E œ ß" Þ# Þ%

Þ" Þ* Þ# Þ" Þ# Þ*

Ô ×Õ Ø

Matrices and linear systems

x bœ œ ßB "B Þ(B #Þ*


"

#

$

then the equation reads:

E œ Þx b

Thus can concisely describe systems usingmatrices.

Matrices and linear systemsGenerally: if we have

+ B + B á + B œ ,"" " "# # "8 8 "

+ B + B á + B œ ,#" " ## # #8 8 #

+ B + B á + B œ ,$" " $# # $8 8 $

ã

+ B + B á + B œ ,7" " 7# # 78 8 7

Matrices and linear systemsCan be described by where:E œx b

E œ œ

+ + á ++ + á +ã ã ã

+ + á +

+


c d"" "# "8

#" ## #8

7" 7# 78

34

x bœ œ

B ,B ,ã ãB ,

Ô × Ô ×Ö Ù Ö ÙÖ Ù Ö ÙÕ Ø Õ Ø

" "

# #

8 7

Matrices and linear systems[now we can concisely write any system ofequations].

Matrix multiplication and vectors7. Matrix multiplication and vectors.

Ex 1: System from above:


" Þ# Þ% B " Þ" Þ* Þ# B Þ( Þ" Þ# Þ* B #Þ*

œ"

#

$

E œ Þx b

General form:

Matrix multiplication and vectors

Ô × Ô × Ô ×Õ Ø Õ Ø Õ Ø+ + + B ,+ + + B ,+ + + B ,

œ"" "# "$ " "

#" ## #$ # #

$" $# $$ $ $

Write:

E œ Ò Óa a a" # $

where

a a a" # $œ à œ à œ Þ" Þ# Þ%

Þ" Þ* Þ#Þ" Þ# Þ*


Matrix multiplication and vectorsFocus on left side:

E œ Ò ÓBBB

x a a a" # $

"

#

$

Ô ×Õ Ø

œ+ + + B+ + + B+ + + B


"" "# "$ "

#" ## #$ #

$" $# $$ $

Matrix multiplication and vectors

œ+ B + B + B+ B + B + B+ B + B + B

Ô ×Õ Ø

"" " "# # "$ $

#" " ## # #$ $

$" " $# # $$ $

œ + B + B + B+ B + B + B+ B + B + B


"" " "# # "$ $

#" " ## # #$ $

$" " $# # $$ $

œ B B B+ + ++ + ++ + +

" # $

"" "# "$

#" ## #$

$" $# $$


Matrix multiplication and vectorsœ B B B" # # $ $a a a"

Conclusion:

E œ Ò Ó œ B B BBBB

x a a a a a a" # $ " # # $ $

"

#

$

Ô ×Õ Ø "

Ex 2:

E œÔ ×Õ Ø

" Þ# Þ% Þ" Þ* Þ# Þ" Þ# Þ*

Þ

Matrix multiplication and vectorsThen

E œxÔ ×Ô ×Õ ØÕ Ø

" Þ# Þ% B Þ" Þ* Þ# B Þ" Þ# Þ* B

"

#

$

œ B B B Þ" Þ# Þ%

Þ" Þ* Þ#Þ" Þ# Þ*

" # $


[another way of writing it]

Matrix multiplication and vectorsNote that is:E œx b

B B B œ Þ" Þ# Þ% "

Þ" Þ* Þ# Þ(Þ" Þ# Þ* #Þ*

" # $

Ô × Ô × Ô × Ô ×Õ Ø Õ Ø Õ Ø Õ Ø

Spanning and equations8. Spanning and equations:Ex 3: Consider question: What is the span of the vectors

a a a" # $œ ß œ ß œ" # !" ! #! " "

Ô × Ô × Ô ×Õ Ø Õ Ø Õ Ø and ?

Answer: Find all vectors obtained as:b œÔ ×Õ Ø,,,

"

#

$

- - - œ" " # # $ $a a a b (1)

for some .- ß - ß -" # $

Spanning and equationsEquations read:

Ò Ó œ- ,- ,- ,

a a a" $

" "

# #

$ $

#


or

E œ E œ Ò Ó œ ß œ- ,- ,- ,

c b a a a c bwhere , " # $

" "

# #

$ $


or

Spanning and equations

Ô ×Ô × Ô ×Õ ØÕ Ø Õ Ø

" # ! - ," ! # - ,! " " - ,

œ Þ" "

# #

$ $

Question: for what is there a solution , ß , ß , - ß - ß -" # $ " # $

above.

[ .]System of 3 eqn's in 3 unknowns

Spanning and equationsAugmented matrix:


" # ! l , " # ! l ," ! # l , ! # # l , ,! " " l , ! " " l ,

Ä" "

# " #

$ $

Ä" # ! l ,! " " l , Î# , Î#! " " l ,

Ô ×Õ Ø

"

" #

$


Ä" # ! l ,! " " l , Î# , Î#! ! ! l , , Î# , Î#

Ô ×Õ Ø

"

" #

$ " #

(2)

Note: exists solution iff , , Î# , Î# œ !Þ$ " #

Conclude solution does NOT exist for all bÞ

Note that:equation (1) has solution for ALL b bÍ −every vector is a linear combination‘$

of a a a" # $ß ß span Í ß ß Þa a a" # $

$‘


[Note also: if left side of had a pivot entry on last row,Ð#Ñwe wouldn't have problem with no solutions].

Conclude: also:

equation (1) has a solution for ALL echelon form of b Í Ehas a pivot entry in every row.

Theorem 4: The following are equivalent:(a) The columns of span a a a"

$ß E# $ß ‘(b) has a solution for each Ex b bœ − ‘$

(c) has a pivot position in every row.E


Proof: In book.

More generally, this theorem holds for any set of vectors:

Theorem 5: The following are equivalent for a matrix7‚ 8E:(a) The columns of a matrix span a a a"

7ß E# 8ßá ß ‘(b) has a solution for each Ex b bœ − ‘8

(c) has a pivot position in every rowE Þ

Spanning and equationsNote this gives an algorithm for checking whether vectorsa a" 8

8ßá ß E span - form matrix and check if every row‘has a pivot position.

[we now start material from section 1.5]

Properties of matrix-vector products:

Theorem 6: If is a matrix and and are vectors and E -u v is a scalar, then:(a) EÐ Ñ œ E Eu v u v(b) EÐ- Ñ œ -EÐ Ñu u .

Proof: Ð+Ñ À

EÐu+v a a aÑ œ Ò á Ó

? @? @

ã? @

" # 8

" "

# #

8 8


œ Ð? @ Ñ Ð? @" " " # #a a aÑ á Ð? @ Ñ# 8 8 8

œ Ð Ð@? á ? Ñ á @ Ñ" " 8 8 " " 8 8a a a a

œ Eu vE Þ

( ) In book - proof is similar.,

Vectors in 2D and 3D Vectors 1. Three dimensional ...math.bu.edu/people/mkon/ma242/L2.pdf ·...

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