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