Planes in Space

1
Planes in Space
2
z
Equation of a Plane
y
x
3
z
Equation of a Plane
y
x
4
z
Equation of a Plane
y
x
5
z
Equation of a Plane
y
x
6
z
Equation of a Plane
y
b
x
7
z
Equation of a Plane
c
y
x
8
z
Equation of a Plane
y
x
9
z
Equation of a Plane
n
P
y
x
10
z
Equation of a Plane
P(x0,y0,z0)
Q(x,y,z)
naibjck
r(x-x0)i(y-y0)j(z-z0)k
n
Q
r
P
r
Q
y
x
11
z
Equation of a Plane
P(x0,y0,z0)
Q(x,y,z)
naibjck
r(x-x0)i(y-y0)j(z-z0)k
n
Q
r
P
r
Q
y
Vector Equation
x
Scalar Equation
12
z
Equation of a Plane
P(x0,y0,z0)
Q(x,y,z)
naibjck
r(x-x0)i(y-y0)j(z-z0)k
n
Q
r
P
r
Q
y
Vector Equation
x
Scalar Equation
13
z
Equation of a Plane
P(x0,y0,z0)
Q(x,y,z)
naibjck
r(x-x0)i(y-y0)j(z-z0)k
n
Q
r
P
r
Q
y
Vector Equation
x
Scalar Equation
14
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.

15
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.

16
Examples
z

• Find the equation of
• the plane pictured.

4
y
5
3
x
17
Graphing Planes

• Sketch the following planes
• (a) 3x - 2y z 6
• (b) z 2y 4
• (c) y 2

18
Angle Between Planes

• Find the angle between the two planes
• 2x 3y 4z 6
• and
• x 2y 3z -1