Cross sections and coordinates

Season 2 • Episode 17 • Cross sections and coordinates

⁰

Cross sections and coordinates

Season 2

Episode 17 Time frame 2 periods

Prerequisites :

^{Crosssetionsofaube.}

Objectives :

•

^{Workin athree} dimensional oordinate.

•

^{Study equations of lines and planes.}

Materials :

•

^{Task sheet.}

•

^{Cut-out ubes.}

•

^Lesson.

•

^Beamer.

1 – Cartesian equations and planes 20 mins

Working in pairs students have to answer a series of questions about some equations of

planes in 3D.

2 – Parametric equations and lines 20 mins

Working in pairs students have to answer a series of questions about some parametri

equations of lines in 3D.

3 – Lesson 15 mins

The teaher explains the main onepts of artesian equations and parametri equations

in 3D. The methods tond intersetions of planes with lines are also introdued.

4 – Using coordinates to find cross sections ^{1 period}

Students are stillworking inpairs. They to draworss setions of the ube usingoordi-

nates. Whiledoing so, they have tomaster the following tehniques :

•

^{nd a artesian equation of aplane knowig three}non-ollinear points;

•

^{nd parametri equations desribing aline;}

•

^{nd the oordinates of the} intersetion of a plane and aline.

Document Task sheet

For this ativity we onsider a ube

OABCDEF G

^{with side}

1cm

^{, as pitured}

on the right.

Using the point

O

^{as the origin and the}

vetors

− − − − − →

OA

^,

− − − − − →

OB

^and

− − − − − →

OC

^{fordiretionsand}

units, we dene a three dimensional arte-

sain system. The position of any point in

the 3D spae, and in partiular any point

ontheube,anbedesribedwithatriplet

ofnumbers

(x, y, z)

^{:theoordinates ofthe}

point. For example, the oordinates of the

point

O

^are

(0 , 0 , 0)

^.

O A

E D

C

F

B G

Preliminary question :

^{Give the oordinates of allthe ube verties inthis system.}

Part 1 – Cartesian equations and planes

1

^{. Find allthe verties ofthe ubesuh that}

z = 0

^{.Doesthis equationsdene aline?}

If not so,what is the set of the pointson the ube suh that

z = 0

^.

2

^{. Desribe inthe same way the set of pointson the ube suhthat}

x = 0

^{and the set}

of points suh that

z = 1

^.

3

^{. Givethe artesian equationsof the 8faes of the ube.}

4

^{. Find all the verties of the ube suh that}

x = y

^{then do a quik sketh of the set}

of the pointsinside the ube suhthat

x = y

^.

5

^{. Same question for the sets suh that}

a

^.

x + y = 1

^;

b

^.

x + y + z = 1

^;

c

^.

x + 2z = 1

^;

d

^.

x + y − z = 1

^;

e

^.

2x + 2y + 2z = 1

^;

f

^.

x + 2y + z = 1

^.

6

^{. What seems to be the general form of a artesian equation of a plane in a 3D}

oordinate system?

7

^.

a

^{. Find a artesian equation of the plane passing through the points}

O

^,

F

^and

N (1 , O, ¹

2 )

^.

b

^{. Findthe oordinatesoftheonlypointonthisplanesuhthat}

x = 0

^and

y = 1

^.

c

^{. Do aquik sketh of that set inthe ube.}

8

^.

a

^{. Find a artesian equation of the plane passing through the points}

A

^,

C

^and

R( ¹ ₂ , O, 1)

^and

F

^.

b

^{. Findthe oordinatesoftheonlypointonthisplanesuhthat}

x = 0

^and

z = 1

^.

c

^{. Do aquik sketh of that set inthe ube.}

Season 2 • Episode 17 • Cross sections and coordinates

²

Part 2 – Parametric equations and lines

1

^.

a

^{. Desribe with a simple sentene the} partiularity of the oordinates of allthe points onthe line

(OA)

^.

b

^{. Translate the properties you found in the previous question into two simple}

equalities.

c

^{. Oneofthe oordinatesmaynot bereferredtoinyour previousanswers. Why?}

2

^.

a

^{. Find threepointsonthe ube suh that}







x = 1 y = 1 z = t

where

t

^{an be any real number.}

b

^{. Whatistheset ofallpointsontheubedenedbytheseparametriequations.}

Draw it.

3

^{. Find a set of parametri equations desribing the lines}

(OC)

^and

(OD)

^{, then the}

lines

(DG)

^and

(F G)

^.

4

^.

a

^{. Find threepointsonthe ube suh that}







x = t y = t z = 0

where

t

^{an be any real number.}

b

^{. Whatistheset ofallpointsontheubedenedbytheseparametriequations.}

Draw it.

5

^.

a

^{. Find two verties of the ube suhthat}







x = 1 − t y = t z = t

where

t

^{an be any real number.}

b

^{. Find the oordinates of the point in this set when}

t = ¹ ₂

^{, then plae it in the}

ube.

c

^{. Whatistheset ofallpointsontheubedenedbytheseparametriequations.}

Draw it.

6

^{. Find out the set of allpoints suhthat}







x = ¹ ₂ y = t z = 1 − t

where

t

^{an beany real number. Drawit.}

Part 3 – Using coordinates to find cross sections

In this part, we name the midpoints of all

the edges of the ube as shown in the pi-

ture below.

1

^{. Give the parametri equations of all}

the edges of the ube. You will use

these equations oftenin this part.

2

^{. In this question we wnat to nd}

the setion of the ube by the plane

(AEL)

^.

a

^{. Findaartesian equationof the}

plane

( AEL )

^.

b

^{. Find the oordinates of the in-}

tersetion of the plane

(AEL)

with eah edge of the plane,

whenever it exists.

c

^{. Plae the points found in the}

previous question and draw the

setionof the ube.

O A

B C

D E

F G

b b

b

b

b b b b b b

b

b

I

J K

L

M N

Q P

R S

T U

3

^{. In this question we wnat tond the setionof the ube by the plane}

(IJU )

^.

a

^{. Find aartesian equation of the plane}

(IJU )

^.

b

^{. Find the oordinates of the} intersetion of the plane

(IJU )

^{with eah edge of}

the plane, whenever itexists.

c

^{. Plae the points found in the previous question and draw the setion of the}

ube.

4

^{. In this question we wnat tond the setionof the ube by the plane}

(IQS)

^.

a

^{. Find aartesian equation of the plane}

(IQS)

^.

b

^{. Find the oordinates of the} intersetion of the plane

(IQS)

^{with eah edge of}

the plane, whenever itexists.

c

^{. Plae the points found in the previous question and draw the setion of the}

ube.

Season 2 • Episode 17 • Cross sections and coordinates

⁴

Document 1

^Cubes

O A

E D

C

F

B G

O A

E D

C

F

B G

O A

E D

C

F

B G

O A

E D

C

F

B G

O A

E D

C

F

B G

O A

E D

C

F

B

G

O A E D

C

F

B G

O A

E D

C

F

B G

O A

E D

C

F

B G

O A

E D

C

F

B G

O A

E D

C

F

B G

O A

E D

C

F

B

G

Season 2 • Episode 17 • Cross sections and coordinates

⁶

O A

E D

C

F

B G

O A

E D

C

F

B G

O A

E D

C

F

B G

O A

E D

C

F

B

G

Document Lesson

1 Cartesian equations and planes

Definition 1

The set of points M(x, y, z) in the 3D space such that ax + by + cz = d where a, b, c and d are real numbers is a plane.

Definition 2

The previous equality, true only for the points on the plane, is a cartesian equation of the plane.

Theorem 1

A plane has infinitely many different cartesian equations.

How to nd a artesian equation

Find aartesian equation of the plane

P

passingthrough the pointsof oor- dinates

P (1, 0, 0)

^,

Q( ¹ ₂ , 1, 0)

^and

R(0, 0, 1)

^.

Weknowthat a artesian equation has the form

ax + by + cz = d.

As the points

P

^,

Q

^,

R

^{are onthe plane,we dedue that}

a

^,

b

^,

c

^and

d

^{are solutionstothe}

system







a × 1 + b × 0 + c × 0 = d a × ¹ ₂ + b × 1 + c × 0 = d a × 0 + b × 0 + c × 1 = d

From the rst equationwe get

a = d

^{,and fromthe last}

c = d

^{.Then, theseond equation}

is equivalentto

1

2 d + b = d

^,or

b = ¹ ₂ d

^.

Asthereareinninitelymanyartesianequation,weanhoosefor

d

^{anynumber(exept}

0

^{inthis ase).Let's hoose}

d = 2

^.

Then we dedue

a = 2

^,

c = 2

^and

b = 1

^{,so a artesian equationof the plane is}

2x + y + 2z = 2.

Season 2 • Episode 17 • Cross sections and coordinates

⁸

2 Parametri equations and lines

Definition 3

The set of points M(x, y, z) in the 3D space such that







x = x _A + at y = y _A + bt z = z _A + ct

where x _A , y _A , z _A , a, b and c are real numbers, is a line passing through point A and with directing vector ~ u(a, b, c).

Definition 4

The previous system, true only for the points on the line, is a set of parametric equations of the line.

Theorem 2

A line has infinitely many different sets of parametric equations.

How to nd parametri equations

Find aset of parametriequationsof the line

L

passingthrough the point of oordinates

T (1, 1, 0)

^{and with diretingvetor}

~ u(0, 0, 1)

^.

Aording to the denition, aset of parametriequations of this line is







x = 1 + 0 × t y = 1 + 0 × t z = 0 + 1 × t

or







x = 1 y = 1 z = t

3 Intersetions of lines and planes

Method

To nd the oordinates of the intersetion of a plane with a line (if it exists), just solve

the system made of a artesian equation of the plane and a set of parametri equations

of the line.

•

^{If the line intersets the plane in one just point, the solving will give one value for}

t

^,

that an be used in the parametri equationsto nd the oordinates of the point.

•

^{If the lineis parallelto the plane,then it willbe impossible tosolve the system.}

•

^{If the lineis inluded inthe plane,then there willbe an innitenumber of solutions.}

An example

Find the oordinates of the intersetion of the plane

P

with the line

L

, if itexists.

Todo so,we have to solve the system :



 



 



2 x + y + 2 z = 2 x = 1 y = 1 z = t

We an replae

x

^,

y

^and

z

^{by their} expressions as a funtion of

t

^{in the rst equation.}

That gives:

2 × 1 + 1 + 2t = 2 3 + 2t = 2

2t = − 1 t = − 1 2

Then, we use this value of

t

^{and the parametri equations tond the oordinates of the}

point:







x = 1 y = 1 z = − ¹ ₂

So the plane and the lineintersetionin the point of oordinates

1, 1, − 1 2

.