Planes in Space. z x y Equation of a Plane z x y.
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Planes in Space
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Transcript of Planes in Space. z x y Equation of a Plane z x y.
Planes in Space
z
x
y
Equation of a Plane
0x
z
x
y
Equation of a Plane
0y
z
x
y
Equation of a Plane
0z
z
x
y
Equation of a Plane
ax
a
z
x
y
Equation of a Plane
by
b
z
x
y
Equation of a Plane
cz
c
z
x
y
Equation of a Plane
z
x
y
Equation of a Plane
P
n
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
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
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
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
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.
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.
Examples
• Find the equation of
the plane pictured.
z
x
y
3
5
4
Graphing Planes
Sketch the following planes:
(a) 3x - 2y + z = 6
(b) z + 2y = 4
(c) y = 2
Angle Between Planes
• Find the angle between the two planes
2x – 3y + 4z = 6
and
x + 2y – 3z = -1
