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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
Direction Cosines
Example
Example
Example
Examples
• Find the equation of
of the line through the
origin and perpendicular
to the plane pictured.
• Find the equation of the
plane perpendicular to
x(t)=4-2t, y(t)= -1+t, z(t)=3
z
x
y
3
5
4
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