Assigned work: pg. 468 #3-8,9c,10,11,13-15 Any vector perpendicular to a plane is a “normal ” to...

Assigned work: pg. 468 #3-8,9c,10,11,13-15

Any vector perpendicular to a plane is a “normal ” to the plane. It can be found by the Cross product of any 2 direction vectors of the plane.

The normal to a plane is used to determine many properties of a plane

8.5 Scalar Equation of a Plane in Space

Proof of Scalar Equation of a Plane:Let be 2 points on the plane.

Let the normal to the plane be 0 0 0 0( , , ) ( , , )P x y z and P x y z

( , , )n A B C

0

0 0 0

0 0 0

0

( , , ) ( , , ) 0

( ) 0

0

n P P

A B C x x y y z z

Ax By Cz Ax By Cz

Ax By Cz D

��


Therefore the Cartesian (Scalar) Equation of a Plane is:

Where : A>0 and A,B,C,D are integers

0Ax By Cz D

8.5 Scalar Equation of a Plane in SpaceEx 1: Find the Cartesian/Scalar equation of the plane:

a)That passes through point (-3,1,-7) and has normal vector (2,4,-5)

0( , , ) ( 3,1, 7) (2,4, 5)P x y z P n

0 0

( 3, 1, 7) (2,4, 5) 0

2 4 5 33 0

P P n

x y z

x y z

��


Ex 1: Find the Cartesian/Scalar equation of the plane:

b) that represents the xz plane.

0

:

(0,0,0)

( , , )

(0,1,0)

NOTE

xz plane contains the origin P

P x y z

any vector along the y axis is

perpendicular to the xz plane

so n j


Similarly equations for:

xy plane: z=0

yz plane: x=0

0 0

( , , ) (0,1,0) 0

0

P P n

x y z

y is the equation for xz plane

��

( (0,0,1))

( (1,0,0))

where n

where n


c) that contains the points A(2,4,-1), B(3,0,2) and C(-1,-2,5).

1

2

:

(3,0,2) (2,4, 1)

(1, 4,3)

( 1, 2,5) (2,4, 1)

( 3, 6,6)

(1,2, 2)

First find direction vectors

d AB

d AC

��

��


1 2

:

(1, 4,3) (1,2, 2)

(2,5,6)

Next find the normal vector

d d

��

:

0

( 2, 4, 1) (2,5,6) 0

2 5 6 18 0

Now find the Scalar equation

AP n

x y z

x y z

��

8.5 Scalar Equation of a Plane in SpaceEx 2: Find the vector and parametric equations of the plane, , parallel

to : and passing through the point B(2,3,-1).

1int :

(0,0,5) (0, 5,0) (5,0,0)

Possible po s on

X Y Z

2 1 : 5x y z

1

2

(0, 5, 5) (0, 1, 1)

(5,0, 5) (1,0, 1)

d XY

d XZ

��

��


Therefore the vector equation is:

The parametric equations are:

(2,3, 1) (0, 1, 1) (1,0, 1)r s t

2 1

3 1

1 1 1

x t

y s

z s t


The Catesian/Scalar equation is used most often because:

1)It is simpler than the vector or parametric forms.

2)Unlike vector or parametric forms, there is only ONE Cartesian/Scalar equation for each plane.



From a Scalar equation we can easily identify the normal.

The normal is often used to:

***Identify whether two planes are parallel, coincident or perpendicular.



Coincident Planes:-Scalar equations of planes are scalar multiples of each other. Ex:

1

2

1 2

: 2 3 1 0

: 4 2 6 2 0

1

2

x y z

x y z

Since planes coincident


Parallel Planes:-When normal vectors are parallel and they don’t share a common point. Ex:

Perpendicular Planes:

-When normal vectors are perpendicular. Ex:

1

2

1 2

1 2

: 2 3 1 0

: 4 2 6 4 0

1

21

2

x y z

x y z

Since n n planes parallel

and

��

1 2 0n n ��


Ex 3: Determine whether the following planes are parallel, coincident, perpendicular or neither.

a) b)

1

2

: 2 3 0

: 0

x y z

x y z

1

2

: 2 2 2 6 0

:3 3 3 9 0

x y z

x y z

1

2

1 2

(2, 1,1)

(1,1, 1)

0

n

n

n n

planes perpendicular

��

��

��

1

2

1 2

1 2

(2, 2, 2)

(3, 3, 3)

2

32

3

n

n

n n but

planes parallel

��

��

��


Normals can also be used to find whether a line is parallel and off, parallel and on or perpendicular to a plane.

Line Perpendicular to a Plane:

Line Parallel to a Plane:

(parallel and on if they ALSO share a common point)

ld kn��

0ld n �� ld kn

��


Ex. 4:

Is the line

parallel to the plane ?

Is so, does it lie on or off the plane?

4 10 0x y z (4,0,3) (1, 2,2),r t t R


But we must now check if the line is on or off the plane.

(1, 2,2) (4,1, 1)

0

sin 0

&

l

l

d n

ce d n

line plane parallel

��

��

(1, 2,2) (4,1, 1)ld n ��


Therefore line is parallel and off the plane.

. . . .

4 10 0

4(4) 0 3 10 3

. . . .

L S R S

x y z

L S R S


Angle between two planes with normals

1 2

1 2

cosn n

n n

��

��1 2n and n��

Assigned work: pg. 468 #3-8,9c,10,11,13-15 Any vector perpendicular to a plane is a “normal ” to...

Documents

Transcript of Assigned work: pg. 468 #3-8,9c,10,11,13-15 Any vector perpendicular to a plane is a “normal ” to...