55 the Vector Equation of a Plane

7/25/2019 55 the Vector Equation of a Plane

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55: The Vector Equation of55: The Vector Equation ofa Planea Plane

Christine Crisp

Teach A Level MathsTeach A Level Maths

Vol. : A Core Mo!ulesVol. : A Core Mo!ules


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The Vector Equation of a Plane

Mo!ule C"

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ME#$%C&


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There are 3for's of the equation of a plane. (eare )oin) to loo* at 2of the'.

+uppose ,e have a vector nthrou)h a point A.

There is onl- oneplane throu)h A that

is perpen!icular tothe vector.

A

n


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There are 3for's of the equation of a plane. (eare )oin) to loo* at 2of the'.

+uppose ,e have a vector nthrou)h a point A.

A

n

R

Then/ 0. =ARn

0)(. =

arn

This is the equation of the plane since it issatisfie! 0- the position vector of an- point onthe plane/ inclu!in) A.

+uppose Ris an- pointon the plane 1 other

than A2.

The scalar pro!uct can 0e epan!e! to )ive

0.. =

anrn anrn ..

=


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#t is useful in so'e pro0le's to *no, that a vectorn,ill 0e perpen!icular to the plane if it is

perpen!icular to 2non3parallel vectors in the plane.

CA

There are an infinite nu'0er of vectors perpen!icular

to AC. 4or ea'ple/ one lies on the plane.


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C

%thers lie at an)les to the plane.

A



%nl- one is also perpen!icular to A.


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C

%thers lie at an)les to the plane.

B



This one is perpen!icular to the plane.%nl- one is also perpen!icular to A.

A


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e.).6 4in! the equation of the plane throu)h the

point A(2, 3, 1) perpen!icular to .+olution:

2

3

1

=

1

32

2

31

2

31

..

z

yx

anrn .. =

292

2

31

.

z

yx

9

2

31

. =

z

yx


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Calculatin) the left3han! scalar pro!uct )ives theCartesian for' of the equation.

923 =

zyx

923

1

. =

zy

x


10/37


e.). +ho, that the vector nis perpen!icular to

the plane containin) the points A/ Ban! C,here

=

6

1

2

n

=

1

3

2

a

2

3

2

b

=

1

1

1

c

+olution: The plane containin) A/ Ban! Calso

contains the vectors an!AB

AC


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=

abAB

=

13

2

23

2

=

acAC =

13

2

11

1

02

1

16

0

=

6

1

2

n

=

13

2

a

2

3

2

b

=

1

1

1

c

,016

0

61

2

.. =

ABn

+o/

002

1

61

2

.. =

ACn

nis perpen!icular to 2vectors in the plane so isperpen!icular to the plane.


12/37


The vector equation of a plane is )iven 0-+7MMA&8

,here ais the position vector of a fie! point onthe plane

nis a vector perpen!icular to the plane an!ris the position vector of an- point on the plane.

nis calle! the nor'alvector

0)(. =

arn or anrn .. =

The Cartesian for' is

dznynxn =321

,here n1/ n2an! n3are the co'ponents of nan!

and .


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Eercise

6. 4in! a vector equation of the plane throu)h

the point ,ith nor'al vector

3

2

1

n

)1,1,1(A

)2,3,4(

A

=

1

20

n

. 4in! the Cartesian equation of the plane throu)hthe point perpen!icular to the vector


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6.

3

2

1

n

Plane throu)h the point ,ith nor'alvector

)2,3,4( A

+olution:

anrn .. =

2

3

4

3

2

1

3

2

1

.. r 8

3

2

1

.

r


15/37


)1,1,1(A

=

1

20

n

. 4in! the Cartesian equation of the plane throu)hthe point perpen!icular to the vector

+olution:

anrn .. =

=

1

1

1

1

2

0

1

2

0

.. r 1

1

2

0

. =

r

1

1

20

. =

z

yx

12 =zy


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+olution:

9. +ho, that is perpen!icular to the plane

=

1

0

3

n

containin) the points A(1, 0, 22/ B(2,3, 1)an! C(2, 2, 1 ).

=

20

1

12

2

=

AC

32

1

=

AB

=

20

1

13

2

33

1

,0

3

3

1

1

0

3

.. =

ABn 0

3

2

1

1

0

3

.. =

=ACn

nis perpen!icular to 2vectors in the plane so isperpen!icular to the plane.

Eercise


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The #ntersection of a Line an! a Plane

#f a line is not parallel to a plane/ it ,ill intersect it.

)22(32: kjitkjirL

e.). 4in! the point of intersection of the line Lan!the plane )iven 0-:

+olution: The point of intersection is the point ,ithvector rsatisf-in) 0oth equations.

Can -ou see ho, to fin! this rA;+: +u0stitute for rfro' Linto . +olve to

fin! tan! su0stitute 0ac* into L.

2)2(.: =

kjir


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=

22

1

13

2

: trL 212

1

.: =

r

#


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Eercise

6. 4in! the point of intersection of the line Lan! the plane ,here

3

4

1

1

2

3

: t

z

y

x

L 33

1

3

2

: . =

z

y

x

. =o, can -ou tell that the follo,in) line Lan!

plane !on


20/37


3

4

1

1

2

3

: t

z

y

x

L 33

1

3

2

: . =

z

y

x

+olution:

6.

+u0s. in

: 33

1

3

2

31

42

3

. =

t

t

t

333112626 =ttt

2211=t

2t


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2t

+u0s. in L:

3

4

1

2

1

2

3

z

y

x

=

7

10

5

z

y

x

Coor!inates are )7,10,5(


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. =o, can -ou tell that the follo,in) line Lan!

plane !onero sho,in)the- are perpen!icular.

+ince 11, 2, 12 is on the line 0ut not onthe plane/ the line an! plane !o not intersect.

A

5

1

2

n

The line an! plane can onl- intersect if the line lieson the plane.


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4in!in) An)les

A

n

the !irection vector of the line an! the nor'al of the plane.

p

np

pn .cos

=

90

Tip: #t


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4in!in) An)les

These lines . . .

The an)le 0et,een planes

are perpen!icular tothe line of intersectionof the planes.

4 ! l


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4in!in) An)les

The qua!rilateral has 2an)les a!!in) to 180

1n


2n

180

so/

4 ! l


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4in!in) An)les

1n


The an)le 0et,een planes is equal to the an)le0et,een the nor'al vectors to the planes.

2n

#


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4in!in) An)les

The an)le 0et,een a line Lan! a plane

is )iven0-

+7MMA&8

nppn .cos

=

90 ,here

is the acutean)le 0et,een the !irection vectorof the line/ / an! the nor'al vector to the

plane/ n.

p

The an)le 0et,een 2planes 1an! 2is the an)le

0et,een their nor'al vectors an! is )iven 0-

21

21.

cosnn

nn

,here n1an! n2are the nor'al vectors to the

planes. #f necessar- su0tract fro' to

fin! the acute an)le.

180

Al,a-s !o a s*etch ,hen fin!in) an)les.An- line or plane1s2 ,ill !o.

4i !i A l


28/37

4in!in) An)les

6. 4in! the an)le/ to the nearest !e)ree/ 0et,een

the line an! plane )iven 0elo,

3

1

0

1

: . =

z

y

x

Eercise

. 4in! the an)le/ to the nearest !e)ree/ 0et,eenthe planes )iven 0elo,

3

0

32

: .2 =

z

yx

4

4

1

1

0

2

: s

z

y

x

L

4

1

14

: .1 =

z

yx

4i !i A l


29/37

4in!in) An)les

6.

44

1

10

2

: szy

x

L 310

1

: . =

zy

x

+olutions:

A

n

p

np

pn .

cos =

90

332

5cos

=

3852

4i !i A l


30/37

4in!in) An)les

4.

1

1

4

:1

z

y

x

.

3

0

3

2

: .2 =

z

y

x

+olutions:

21

21 .

cos nn

nn

1318

5cos

109

71Acute an)le is

1n

2n

Th V t E ti f Pl


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Th V t E ti f Pl


32/37


The follo,in) sli!es contain repeats ofinfor'ation on earlier sli!es/ sho,n ,ithoutcolour/ so that the- can 0e printe! an!

photocopie!.

4or 'ost purposes the sli!es can 0e printe!as =an!outs ,ith up to 6sli!es per sheet.

Th V t E ti f Pl


33/37


The vector equation of a plane is )iven 0-+7MMA&8

,here ais the position vector of a fie! point onthe plane

nis a vector perpen!icular to the plane an!ris the position vector of an- point on the plane.

nis calle! the nor'al vector

0)(. =

arn or anrn .. =

The Cartesian for' is

dznynxn =321

,here n1/ n2an! n3are the co'ponents of nan!

and .

Th V ct r Equ ti n f Pl n


34/37


e.).6 4in! the equation of the plane throu)h the

point A(2, 3, 1) perpen!icular to .

+olution:

23

1

=

13

2

23

1

23

1

..zy

xanrn ..

=

292

2

3

1

.

z

y

x

9

2

3

1

. =

z

y

x

Calculatin) the left3han! scalar pro!uct )ives theCartesian for' of the equation.

923 =

zyx



35/37


A

n

The An)le 0et,een a Line an! a Plane

the !irection vector of the line an! the nor'al of the plane.

p

np

pn .cos

=

90

Tip: #t


36/37


1n


2n

#


37/37


The an)le 0et,een a line Lan! a plane

is )iven0-

+7MMA&8

np

pn .cos

=

90 ,here

is the acute an)le 0et,een the !irection vectorof the line/ / an! the nor'al vector to the

plane/ n.

p

21

21 .

cos nn

nn

#f necessar- su0tract fro' to fin! acute an)le.

180

Al,a-s !o a s*etch ,hen fin!in) an)les.

A li l 1 2 ill !

,here n1an! n2are the

nor'al vectors to the planes.

The an)le 0et,een 2planes 1an! 2is the an)le

0et,een their nor'al vectors an! is )iven 0-

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55 the Vector Equation of a Plane