lines in space
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Lines in Space
z
x
y
P
Q
Equation of a Line
z
x
y
r0
dP
Q
Equation of a Line
z
x
y
r0
d
rP
Q
Q’
Equation of a Line
z
x
y
r0
d
rP
Q
Q’
P(x0,y0,z0)
Q(x1,y1,z1)
Q’(x,y,z)
d=d1 i+d2 j+d3 k
r0=x0 i+y0 j+z0 k
=(x1 -x0)i+(y1-y0)j+(z1-z0)k
Equation of a Line
z
x
y
r0
d
rP
Q
Q’
P(x0,y0,z0)
Q(x1,y1,z1)
Q’(x,y,z)
Vector Parameterization
d=d1 i+d2 j+d3 k
r0=x0 i+y0 j+z0 k
=(x1 -x0)i+(y1-y0)j+(z1-z0)k
Equation of a Line
z
x
y
r0
d
rP
Q
Q’
P(x0,y0,z0)
Q(x1,y1,z1)
Q’(x,y,z)
kdjdidtkzjyix
tdrtr
321000
0
)(
Vector Parameterization
d=d1 i+d2 j+d3 k
r0=x0 i+y0 j+z0 k
=(x1 -x0)i+(y1-y0)j+(z1-z0)k
Equation of a Line
z
x
y
r0
d
rP
Q
Q’
P(x0,y0,z0)
Q(x1,y1,z1)
Q’(x,y,z)
kdjdidtkzjyix
tdrtr
321000
0
)(
ktdzjtdyitdx 302010
Vector Parameterization
d=d1 i+d2 j+d3 k
r0=x0 i+y0 j+z0 k
=(x1 -x0)i+(y1-y0)j+(z1-z0)k
Equation of a Line
z
x
y
r0
d
rP
Q
Q’
P(x0,y0,z0)
Q(x1,y1,z1)
Q’(x,y,z)
kdjdidtkzjyix
tdrtr
321000
0
)(
ktdzjtdyitdx 302010
10 tdxtx )( 20 tdyty )( 30 tdztz )(Scalar Parametric Equations
Vector Parameterization
d=d1 i+d2 j+d3 k
r0=x0 i+y0 j+z0 k
=(x1 -x0)i+(y1-y0)j+(z1-z0)k
Equation of a Line
Representations of a Line
Examples
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.
z
x
y
QP
R
cvbuavur
PRvPQuOPOX
),(
Parametric Equation
Parametric Equation of a Plane
P
X
a b
c
z
x
y
QP
R
cvbuavur
PRvPQuOPOX
),(
Parametric Equation
Parametric Equation of a Plane
P
X
a b
c
z
x
y
QP
R
cvbuavur
PRvPQuOPOX
),(
Parametric Equation
Parametric Equation of a Plane
P
X
a b
c
Representations of a Plane
cvbuavur
),(
Parametric Equation
0)()()(000
zzcyybxxa
Scalar Equation
Applications
Angle Between Planes
• Find the angle between the two planes
2x – 3y + 4z = 6
and
x + 2y – 3z = -1
Example
Example
Graphing Planes
• 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
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