2_intersections_of_planes_in_3_dimensions_ver_2.docx

7/24/2019 2_intersections_of_planes_in_3_dimensions_ver_2.docx

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Applying systems of simultaneous equations

in solving problems.

INTERSECTIONS OF PANES in ! "I#ENSIONS.


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1. We are all familiar with equations likey = 2x + 3 or in generaly = mx + c.

We know that such equations represent INESin thex, yplane.

In this case, the gradient is 2 and theyintercept is 3 and we can easily

visualisethe position of any such line.

Line equations are sometimes written in the form ax + by = c in which casethe aboe line would become 2x y = 3. !his of course is the same line with

gradient 2 andyintercept 3.

"oweer, equations such asx + y + z =3 do #$! represent INESin 3

dimensional space. !hey actually represent PANES.

If we choose some points (x, y, z)which fit this equation we obiously get the

following% &3, ', '(, &', 3, '(, &', ', 3(, &1, 1, 1(, &1, 2, '(, &1, ', 2( etc

If we were to plot these we would get the following plane%

&', 3, '(

&1, 1, 1(

&', ', 3( &2, 1, '(

&3, ', '(

2. It is quite difficult to imagine the position of a plane such as 2x + 3y + z = 6

) simple way to do this is to fin$ %&ere t&e plane %oul$ 'ross ea'& a(is.

If we putx = 0 and y = 0 then z = 6so it crosses theza*is at &', ', +(

If we puty = 0 and z = 0 then 2x = 6 so x = 3crossing thexa*is at &3, ', '(

If we putx = 0 and z = 0 then 3y = 6 so y = 2crossing theya*is at &', 2, '(

rossesy a*is

at &', 2, '(

rossesxa*is

at &3, ', '(

rossesza*is

at &', ', +(

3. If we consider the equationx + y z = 3


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---------------------------------------------------------------

. !he plane 4x 3y 2z = 12crosses the a*es at &3, ', '(, &', /, '(, &', ', /+(

z axis at (0, 0, -6)

xaxis at (3, 0, 0)

y axis at (0, -4, 0)

#ost of t&e time) it is not &elpful to %orry about trying to visualise t&e

position of a plane from its equation.

T&e A*TO+RAP, program is available to over'ome t&is $iffi'ulty.

0.#aturally, an equation withx, yandzsuch asx + y + z = 6inoles

uttingy and z = 0 we et x = 3so

it crossesxa*is at &3, ' ,'(

uttingx and z = 0 we et y = 3soit crossesya*is at &', 3, '(

ut puttingx and y = 0 we et z = -3so

it crossesza*is at &', ', /3(

T&is is obviously far more $iffi'ult to

visualisesin'e %e nee$ to e(ten$ t&ez

a(is to -!.


4/16

coordinates such as &3, 1, 2( and clearly represents a PANEin 3/ space

but an equation such asx + y = 6does not mentionzand although it normally

4ust represents a INEin thex, yplane, it can also represent a PANEin the

x, y,z coordinate system.

Ifx + y = 6 thenxandyalues are connected butz can hae )#5 alue.

+. 6imilarly the equationx = 3CO*"represent% &i( a single pointon anxnumber line.

x = 3x ! 4 3 2 1 0

&ii( a verti'al linein thex, yplane.

x = 3

&iii( a verti'al planeinx, y, zspace.

x = 3

Te'&nique for fin$ing t&e interse'tion point of ! planes.

)boex + y = 6is 4ust a INE

in the 2/x, yplane.

)boex + y = 6is a PANE

in 3/ x, y, zspace.


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6/16

In the aboe problem we added equationsand

x + 2y + z = 14

2x + 2y z = 10

7and obtained the equation 3x + 4y = 24

which can be written as y = 3 x + 64

It is a 'ommon mis'on'eption t&at t&is equation represents t&e INE of

interse'tion of t&e planes. IT "OES NOT.

!he actual line of intersection could be written parametrically as%

x = ' 't, y = 6t, z = 6 4t which is beyond the scope of this course.

e %ill NOT use E/*ATIONS of INES in !-"imensional spa'e at all.

It is better to t&in0 ofy = 3 x + 6 as a * on the %ont x, y &ane,

4

%it& a gra$ient of 3 an$ yinter'ept of 6. 4

&i( !his diagram shows the

greeninterse'tion lineof the

two planes%

&ii( !aking away the planes and 4ust

leaing the greeninterse'tionINE%

&iii( If we now drawy = 3 x + 6 on the frontx, yplane &orangeline(

4

we see that it is the pro1e'tion

&or shadow( of the actual line of

intersection of the planes, onto the

x, y plane.&(7and when we added equationsandwe got the equation

3x + y = 1! o% y = 3x + 1!

)gain this is notthe line of

intersection of planesand

)s before, we could say that

y = 3x + 1!

is 4ust a line in thex, yplane with a

gradient 3 andyintercept of 1!.

&i( 6imilarly the T*R/*OISEline

is the intersection of planesand


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!his means that, instead of using the a'tual lines of intersection of the planes,we used the t%o pro1e'te$ lines of interse'tion on the x, y planeto find thex

andycoordinates of the intersection of the three planes.x = 4 and y = 3

Finally %e substitute$ t&ese values into one of t&e plane equations to fin$

t&ezvalue.x = 4, y = 3 and z = 4

2. 8ind the intersection point of the 3 planes gien by these equations%

&ii( rawing both pro4ections of theactual intersection lines of the planes,

we see that 9 is also the pro4ection of

which is the point of intersection of

the planes.

&i( If we now draw this 2*Eliney = -3x + 1!on thex, yplane, we can

see that it is the pro4ection of the

T*R/*OISEline on thex, yplane.

9


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x + y + z = '

x + y z = 4

2x y + z = 2

)dding equationsandwe get 2x + 2y = 12

6ubtracting equationfromwe get x 2y = 6

)ddingandwe get 3x = 6

so x = 2

6ub in 4 + 2y = 12

2y = '

so y = 4

and s#bs inwe et

2 + 4 + z = ' so z = 2

$he inte%section &oint is (2, 4, 2)

!he intersection point can be seen from this 3/ graph%


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10/16

. 6ometimes we hae three equations to sole when at least one of them only

has two ariables.

3x + 2y + z = 13

2x + !y 2z = "

4x + 3y = 13

!his situation makes the problem easier because we only hae to eliminatez

on'einstead of t%i'e.

:liminatingzfromand

2 6x + 4y + 2z = 26

2x + !y 2z = "

+ 'x + "y = 3!

3 12x + "y = 3"

- 4x = 4

o x = 1

#b in 4 + 3y = 13

3y = "

y = 3

s#b in 3 + 6 + z = 13

z = 4

the inte%section &oint is (1, 3, 4)


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,EN PANES +O RON+3

In 4-"imensional spa'e)most lines interse't %it& ea'& ot&er5

eg y = 3x 4 and y = x + 6

we t 3x 4 = x + 6 and soe 2x = 10

x = ! and y = 11

the ines inte%sect at (!, 11)

ut we find that sometimes lines do not intersect.

!his onlyoccurs when the lines are parallel.

eg y = 2x + 6 and y = 2x + 10

we t%y to o th%o#h the aeb%aic &%ocess as beo%e, we et e#ationswhich hae no so#tion.

2x + 6 = 2x + 10

$his eads to no sensibe conc#sion beca#se i we s#bt%act 2x %om both sides

we et 6 = 10.

!hese equations are called inconsistent.

6imilarly, there are seeral cases when 3 planes do not intersect.

1. !he simplest case is for ! parallel planes.

eg x + y + z = 3 x + y + z = 4 $he%e is no common so#tion.

x + y + z = !

$biously, equations of parallel plane hae the SA#E COEFFICIENTS.

)s aboe, the algebra gies us

impossible situations.

If we subtract equationfromwe get ' ; 1

!hese equations are also called

*5*$*$.

arallel planes are easy to pick out

eg 2x + 3y 4z = !

2x + 3y 4z = 11

4x + 6y 'z = 13

#otice is 4ust multiplied by 2.

It could hae been written as %

2x + 3y 4z = 6.!


12/16

2.&a( If 2 of the planes are parallel and the 3rdplane crosses them both, there is

no intersection of all 3 planes.

eg x + y + z = 3

x + y + z = !

3x 2y z = 1

----------------------------------------------------------------

2&b( onsider these 3 planes%

2x + 3y z = 6

2x + 3y z = 14

x + y + z = '

learly planesandare parallel because they hae the same 'oeffi'ientsof

x, y andz.!he plane would cross them both as shown below%

)gain it is easy to pick out thatand

are parallel because the coefficients

ofx, y andzare the same.

If we 6attempt7the usual technique for

trying to find a unique solution we get

impossible situations again.

eg addingand 4x y = 4

and addingand 4x y = 6

$hese 77 ine %a&hs a%e the &%o8ections

on the x,y &ane o the act#a ines o

inte%section o the &anes9and9%es&ectiey.

T&ese equations&ave no solutionand

are also called*5*$*$.

)gain it is easy to pick out thatand

are parallel because the coefficients ofx, yandzare the same.

If we 6attempt7the usual technique fortrying to find a unique solution we get

impossible situations again.

eg addingand 3x + 4y = 14

and addingand 3x + 4y = 22

$hese 77 ine %a&hs a%e the &%o8ections

on the x,y &ane o the act#a ines o

inte%section o the &anes9

and9%es&ectiey.

T&ese equations &ave no solutionandare also called*5*$*$.


13/16

3. &) ery odd situation(%

onsider these equations%

x 2y + z = 1

2x + 3y + z = 12 3x + y + 2z = '

)ttempting a solution in the normal way%

im z %omand

- x + !y = 11

im z %omand

2 2x 4y + 2z = 2

3x + y + 2z = '

- x + !y = 6

im z %omand

2 4x + 6y + 2z = 24

3x + y + 2z = '

- x + !y = 16

Note5 Remember) equations)an$are best 'onsi$ere$ to be !

parallel lines in t&ex, yplane %&i'& are t&e pro1e'tions of t&e a'tual linesof interse'tion of t&e ! planes.

learly the equations,andhae the same coefficients ofxandybut

different constant terms.

Looking at the picture, we see that the 3 planes do not intersect in a single point

but they intersect forming a PRIS#shape.

+ENERA8% !his always occurs if a linear combinationof 4 of t&e

equationsproduces an equation with the same 'oeffi'ientsofx, y andz as the

3rdequation but with a different'onstant term.

x 2y + z = 1

2x + 3y + z = 12

adding 3x + y + 2z = 13 i.e. same coeicients asb#t constant = 13 not

'.

T&ese equations &ave no solutionand are also called*5*$*$.


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15/16

so (-4, 3, 11) is anothe% so#tion.

$he act#a e#ation o% the ine o inte%section o the th%ee &anes is

x = 1 + !t, y = 2 t, z = 4 /t

S*##AR85 +EO#ETRICA INTERPRETATION.1. If the equations are clearly IN"EPEN"ENT, meaning we could not

combine two of the equations and produce the 3rd, then there will be a

*NI/*E SO*TIONwhere the planes cross at a single point (x, y, z)

e x + y + z = 3

2x + 3y + 4z = "

3x + !y + 3z = 11

2. If the equations all hae the same coefficients ofx, y andzbut differentconstant terms, then they are t&ree parallel planesand there is no intersection.

!here are no common points of intersection which means there are no solutions.

($hese e#ations a%e caed inconsistent.)

e 2x + 3y + 6z = !

2x + 3y + 6z = "

2x + 3y + 6z = 12

3. If !W$ of the planes can be shown to hae the same coefficients ofx, y and

zbut different constant terms, then the t%o planes are paralleland the thirdplane cuts each of them in two parallel lines.



e 2x + 3y + 6z = !

2x + 3y + 6z = "

!x + /y + 2z = 12

. If a linear combination of two of the equations has the same coefficients ofx,

y andzas the third equation but the constant term is different, the three planes

form a triangular =I6>.



e 2x + 3y + 6z = !

4x + /y + 2z = "

6x +10y + 'z = 4

0. If a linear combination of two of the equations equals the third equation then

the three planes intersect in the same line and t&ere are infinitely manysolutions.;Ea'& point on t&e line is a solution


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($hese e#ations a%e caed de&endent.)

e 2x + 3y + 6z = !

4x + /y + 2z = "

6x +10y + 'z = 14

2_intersections_of_planes_in_3_dimensions_ver_2.docx

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