Unit7 - Lines and Planes in 3-D Linear Equations: In 2-D:...

Lines and Planes in 3-D
Linear Equations:
In 2-D:
Ax + By =
What is the graph?
C
D
What is the graph?
Review Parametric Equations
General 2-D: Example:
x = f ( t )
y = g ( t )
x = t
2
y = 4 t
2
What about 3-D?
General 3-D: Example:
x = f ( t )
y = g ( t )
z = h ( t )
x = t
2
y = 4 t
2
z = t
2
+ 1

Lines in 3-D
Given: P
1
( x
1
, y
1
, z
1
) is a specific point on the line

v = a , b , c is a vector parallel to the line
Find: Parametric Equations to describe all of the points
P ( x , y , z ) on the line

Parametric Equations for the line that passes through the point
P ( x
1
, y
1
, z
1
)
and is parallel to the vector

v = a , b , c
:
x = x
1
+ at
y = y
1
+ bt
z = z
1
+ ct
___________________________________________________
Find parametric equations for the line that passes through
P
1
(4, − 1,3)
and is parallel to

v = 2,1, − 4
.
Does the point
Q (6,0, − 1)
lie on the line?
Does the point
R (1,2, − 3)
lie on the line?
Is the line parallel to the vector

w = − 4, − 2,8
?
Is the line parallel to the vector

w = 6,3, − 4
?