Planes in Space. z x y Equation of a Plane z x y.
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Transcript of Planes in Space. z x y Equation of a Plane z x y.
![Page 1: Planes in Space. z x y Equation of a Plane z x y.](https://reader035.fdocuments.in/reader035/viewer/2022062221/56649d385503460f94a10b8a/html5/thumbnails/1.jpg)
Planes in Space
![Page 2: Planes in Space. z x y Equation of a Plane z x y.](https://reader035.fdocuments.in/reader035/viewer/2022062221/56649d385503460f94a10b8a/html5/thumbnails/2.jpg)
z
x
y
Equation of a Plane
0x
![Page 3: Planes in Space. z x y Equation of a Plane z x y.](https://reader035.fdocuments.in/reader035/viewer/2022062221/56649d385503460f94a10b8a/html5/thumbnails/3.jpg)
z
x
y
Equation of a Plane
0y
![Page 4: Planes in Space. z x y Equation of a Plane z x y.](https://reader035.fdocuments.in/reader035/viewer/2022062221/56649d385503460f94a10b8a/html5/thumbnails/4.jpg)
z
x
y
Equation of a Plane
0z
![Page 5: Planes in Space. z x y Equation of a Plane z x y.](https://reader035.fdocuments.in/reader035/viewer/2022062221/56649d385503460f94a10b8a/html5/thumbnails/5.jpg)
z
x
y
Equation of a Plane
ax
a
![Page 6: Planes in Space. z x y Equation of a Plane z x y.](https://reader035.fdocuments.in/reader035/viewer/2022062221/56649d385503460f94a10b8a/html5/thumbnails/6.jpg)
z
x
y
Equation of a Plane
by
b
![Page 7: Planes in Space. z x y Equation of a Plane z x y.](https://reader035.fdocuments.in/reader035/viewer/2022062221/56649d385503460f94a10b8a/html5/thumbnails/7.jpg)
z
x
y
Equation of a Plane
cz
c
![Page 8: Planes in Space. z x y Equation of a Plane z x y.](https://reader035.fdocuments.in/reader035/viewer/2022062221/56649d385503460f94a10b8a/html5/thumbnails/8.jpg)
z
x
y
Equation of a Plane
![Page 9: Planes in Space. z x y Equation of a Plane z x y.](https://reader035.fdocuments.in/reader035/viewer/2022062221/56649d385503460f94a10b8a/html5/thumbnails/9.jpg)
z
x
y
Equation of a Plane
P
n
![Page 10: Planes in Space. z x y Equation of a Plane z x y.](https://reader035.fdocuments.in/reader035/viewer/2022062221/56649d385503460f94a10b8a/html5/thumbnails/10.jpg)
z
x
yQ
P
n
r
r Q
Q(x,y,z)
P(x0,y0,z0)
n=ai+bj+ck
r=(x-x0)i+(y-y0)j+(z-z0)k
Equation of a Plane
![Page 11: Planes in Space. z x y Equation of a Plane z x y.](https://reader035.fdocuments.in/reader035/viewer/2022062221/56649d385503460f94a10b8a/html5/thumbnails/11.jpg)
z
x
yQ
P
n
r
r Q
Q(x,y,z)
P(x0,y0,z0)
n=ai+bj+ck
r=(x-x0)i+(y-y0)j+(z-z0)k
0
0
000
000
000
)()()(
)()()(
)()()(
zzcyybxxa
zzcyybxxa
kzzjyyixxckbjai
rn
Scalar Equation
Vector Equation
Equation of a Plane
![Page 12: Planes in Space. z x y Equation of a Plane z x y.](https://reader035.fdocuments.in/reader035/viewer/2022062221/56649d385503460f94a10b8a/html5/thumbnails/12.jpg)
z
x
yQ
P
n
r
r Q
Q(x,y,z)
P(x0,y0,z0)
n=ai+bj+ck
r=(x-x0)i+(y-y0)j+(z-z0)k
0
0
000
000
000
)()()(
)()()(
)()()(
zzcyybxxa
zzcyybxxa
kzzjyyixxckbjai
rn
Scalar Equation
Vector Equation
Equation of a Plane
![Page 13: Planes in Space. z x y Equation of a Plane z x y.](https://reader035.fdocuments.in/reader035/viewer/2022062221/56649d385503460f94a10b8a/html5/thumbnails/13.jpg)
z
x
yQ
P
n
r
r Q
Q(x,y,z)
P(x0,y0,z0)
n=ai+bj+ck
r=(x-x0)i+(y-y0)j+(z-z0)k
0
0
000
000
000
)()()(
)()()(
)()()(
zzcyybxxa
zzcyybxxa
kzzjyyixxckbjai
rn
Scalar Equation
Vector Equation
Equation of a Plane
![Page 14: Planes in Space. z x y Equation of a Plane z x y.](https://reader035.fdocuments.in/reader035/viewer/2022062221/56649d385503460f94a10b8a/html5/thumbnails/14.jpg)
Examples
• Find the equation of the plane through (1,1,2), (3,2,-1) and (4,2,-1).
• Find the equation of the plane through (2,-1,3) and parallel to 3x – y + 4z =12.
![Page 15: Planes in Space. z x y Equation of a Plane z x y.](https://reader035.fdocuments.in/reader035/viewer/2022062221/56649d385503460f94a10b8a/html5/thumbnails/15.jpg)
Intercepts of a Plane
• Find the intercepts of the planes
2x – 3y + z = 6
4y + 2x = 8
z = 3• Sketch the planes.• Find the normals to the planes.
![Page 16: Planes in Space. z x y Equation of a Plane z x y.](https://reader035.fdocuments.in/reader035/viewer/2022062221/56649d385503460f94a10b8a/html5/thumbnails/16.jpg)
Examples
• Find the equation of
the plane pictured.
z
x
y
3
5
4
![Page 17: Planes in Space. z x y Equation of a Plane z x y.](https://reader035.fdocuments.in/reader035/viewer/2022062221/56649d385503460f94a10b8a/html5/thumbnails/17.jpg)
Graphing Planes
Sketch the following planes:
(a) 3x - 2y + z = 6
(b) z + 2y = 4
(c) y = 2
![Page 18: Planes in Space. z x y Equation of a Plane z x y.](https://reader035.fdocuments.in/reader035/viewer/2022062221/56649d385503460f94a10b8a/html5/thumbnails/18.jpg)
Angle Between Planes
• Find the angle between the two planes
2x – 3y + 4z = 6
and
x + 2y – 3z = -1