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5.3

LINES and PLANESin

3R

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z axis

xy-plane:xz-plane:

yz-plane:

y axis

x axis

Planes

0y0

x

0z2These lecture slides were created

by Prof. Babierra.

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Planes

A plane can be uniquely

determined by any of thefollowing:

three non-collinear points

a line and a point not onthe line

two lines with one point ofintersection

two parallel lines

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Planes

x

y

z

333 z,y,x

222 z,y,x

111 z,y,x

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Planes

If is a given non-zero vectorand is a point,N

0P then the set

of all points for whichand are orthogonal is aPLANE

P

PP0N through and having

as a normal vector.0P N

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Planes

Vector:N

PP P

P0PPoint:

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Equation of a plane in 3D

Point on the plane: 0000 z,y,xP

c,b,aNNormal vector to the plane:

0000 zzcyybxxaStandard equation of the plane:

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Equation of a plane in 3D

0 dczbyaxGeneral equation of a plane:

ifa, b and c are not all zero,

is a normal vector tothe planec,b,a

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Remark

Two planes are parallel if theirnormal vectors are parallel, i.e.they are scalar multiples.

Two planes are perpendicular iftheir normal vectors are

orthogonal, i.e. a dot product of0 (zero).

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Example. Determine the equationof the given plane.

1. plane through the pointand perpendicular

to the vector

312

,,

324 ,,

Solution:

312 ,,N

3240

,,P

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Solution (continued)

312

,,N324

0

,,P

0000 zzcyybxxa

0332 zyx

0332142 zyx

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Graph

0332 zyx

1001 ,,P 0302 ,,P

0123 ,,P 1 2 3 4 51

2

34

5

-1-2-3-4-5-1

-2

-3

-4

-5

1

23

45

-2-3

-4-5

x

y

z

3240 ,,P

0P12These lecture slides were created

by Prof. Babierra.

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Example. Determine the equationof the given plane.

2. plane through the pointand parallel to the

plane

0532

zyx:

471 ,,

Solution:

4710 ,,P

MNN 312 ,,13These lecture slides were created

by Prof. Babierra.

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Solution (continued)

312

,,N471

0 ,,P

0000 zzcyybxxa

0332 zyx

0437112 zyx

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Graph

0332 zyx

x-intercept:

y-intercept:

z-intercept:

00 z;y

00 z;x

00 y;x

2

3

3

1

00

23

1 ,,P

0302 ,,P

1003 ,,P

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1 2

1

2

-1-2

-1

-2

1

2

-1

-2

y

zGraph

0332 zyx

312 ,,N K

0532 zyx:16These lecture slides were created

by Prof. Babierra.

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Example. Determine the equationof the given plane.

Solution:

3. plane containing thepoints ,and

032 ,,P 150 ,,Q

301,,R

normal vector ? ? ?

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1 2 3 4 5

1

2

3

4

5

-1-2-3-4-5-1

-2

-3

-4

-5

12

34

5

-2-3

-4

-5

x

y

zSolution N032 ,,P

150 ,,Q301 ,,R

is a vector perpendicular

to , and

NPQ

PRQR

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Solution (continued)

032 ,,P 150 ,,Q 301 ,,R

is a vector perpendicular

to , and

N

PQ

PR QR

122

,,PQ 331 ,,RP

PRPQN

331

122

kji

873 ,,

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Solution (continued)

0000 zzcyybxxa

027873 zyx

083723 zyx

032

,,P873

,,

N

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Equation of a plane in 3D

Point on the plane: 0000 z,y,xP

c,b,aNNormal vector to the plane:

0000 zzcyybxxaStandard equation of the plane:

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Lines in 2D

11 y,x

22 y,x

112

121 xxxx

yyyy

Two-point form

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Lines in 2D

bmxy

Point-slope form 11 xxmyy

Slope-intercept form

1b

y

a

xIntercept form

General equation 0 CByAx

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Lines in 3D

If is a given non-zero vectorand is a point,R

0P then the set

of all points for whichis parallel to is a LINE

P

PP0R

through and parallel to .0P R

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x

y

zLines in 3D

0P

R

P

P

P

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Lines in 3D

Using as a parameter,t

Let be a line that containsthe point and isparallel to the vector

.

L0000 z,y,xP

c,b,aR

atxx 0 btyy 0 ctzz 0

PARAMETRIC EQUATIONS ofL

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Lines in 3D

Let be a line that containsthe point and isparallel to the vector

.

L0000 z,y,xP

c,b,aR SYMMETRIC EQUATIONS ofL

c

zz

b

yy

a

xx 000

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Example. Determine the parametricand symmetric equations of the given

line.1. line through the point

and is parallel tothe vector

L

542 ,,321 ,,

Solution:

542 ,,R 3210 ,,P

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Solution (continued)

atxx 0 btyy 0 ctzz 0

PARAMETRIC EQUATIONS ofL

542 ,,R 3210 ,,P

tx21

ty42

tz53

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Solution (continued)

SYMMETRIC EQUATIONS of L

542 ,,R 3210 ,,P

c

zz

b

yy

a

xx 000

5

3

4

2

2

1

zyx

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Graph

1 2 3 4

1

23

4

5

-1-2-3-4-5-1

-2

-3

-

12

34 5

-2-3

-4-5

y

z

0P

R

tx 21

ty 42

tz 53

542 ,,R 3210 ,,P

L

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Using parametric equations

tx 21 ty 42 tz 53

At ,0t 1x 2y 3z 321 ,,

At ,2t 3x 6y 13z 1363 ,,

At ,1t 3x 6y 2z

263

,,At ,

21

t 1x 0y21101 ,,

211z

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Example. Determine the parametricand symmetric equations of the given

line.2. line through the points

andM

154 ,,Q032 ,,P

Solution:

is parallel to vector .M

PQ122

,,PQ

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Solution (continued)

PARAMETRIC EQUATIONS ofM

tx 24 ty 25 tz 11

154 ,,Q

122

,,PQ

SYMMETRIC EQUATIONS ofM

1

1

2

5

2

4

zyx

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Graph

1 2 3 4 5

1

2

3

4

5

-1-2-3-4-5-1

-2

-3

-4

-5

1 2

34

5

-2

-3-4-5

x

y

z

P Q

M154 ,,Q

122

,,PQ

032 ,,P

PQ

tx 24ty 25

tz1135These lecture slides were createdby Prof. Babierra.

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END

36These lecture slides were createdf