math 37 unit 5.3
TRANSCRIPT
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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
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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
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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
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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
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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
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