Lines and Planes in 3-D
Linear Equations:
In 2-D:
Ax + By = C
What is the graph?
In 3-D: Ax + By + Cz = D
What is the graph?
Review Parametric Equations
General 2-D:
x = f (t)
y = g(t)
What about 3-D?
General 3-D:
x = f (t)
y = g(t)
z = h(t)
Example:
x = t2
y = 4t 2
Example:
x = t2
y = 4t 2
z = t2 +1
Lines in 3-D
Given:
P1 (x1 , y1 , z1 ) 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

v
P(x1, y1,z1 ) and is parallel to the vector = a,b,c :
x = x1 + at
y = y1 + bt
z = z1 + ct
___________________________________________________
Find parametric equations for the line that passes through

P1 (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 ?
Find parametric equations for the line that passes through
the points P(2, 3, −2) and Q(−3, 0, 5) .
Are there other possibilities?
Examine the line: x = 4 − t,
y = 2, z = t
1. Find two vectors parallel to the line.
2. Find two points on the line.
3. Sketch the line.
Intersecting Lines
Examine the two lines:
L1 : x1 = 3 + 5t, y1 = 2 − 5t, z1 = 10t
L2 : x2 = 2 + s, y2 = −7 + 3s, z2 = 8 − 2s
Are they parallel?
Do they intersect?
What about……
L1 : x1 = 3 + 5t, y1 = 2 − 5t, z1 = 10t
L2 : x2 = 2 + s, y2 = −7 + 3s, z2 = 4 − 2s
Are they parallel?
Do they intersect?
Planes in 3-D
Given:
P1 (x1 , y1 , z1 ) is a specific point on the plane

n = a,b, c is a vector perpendicular to the plane
Find:
A linear equation to describe all of the points P(x, y, z) on
the plane
Linear Equation for the plane that passes through the point P1 (x1, y1,z1 )

and is perpendicular (normal) to the vector n = a,b,c :
a(x − x1 ) + b(y − y1 ) + c(z − z1 ) = 0
____________________________________________________
Find the equation for the plane that passes through P1 (4, −1, 3)

and has normal vector v = 2,1, −4 .
Does the point Q(6, 0, −1) lie on the plane?
Does the point R(−2, −13, −3) lie on the plane?
Find the equation for the plane that contains the points
P(1,1, 6) Q(2, −3,1) R(4,1, −2)
Two ways to write the equation of a plane:
1. To find the equation when you know information about the
plane, use the dot product form:
2. To find information about the plane when you know the
equation, use the general linear form:
Find the equation of the plane that contains the point
P(1, 2, 3) and is parallel to the plane 2x − 4 y + z = 10 .
Find parametric equations for the line that contains the point
P(1, 2, 3) and is perpendicular(normal) to the plane
2x − 4 y + z = 10 .
Find the equation of the plane that contains the point
P(1, 2, 3) and is perpendicular to the line with parametric
equations x = 4 − t, y = 2 + 2t, z = 5 + t .
Find the distance from the point P(1, 2, 3) to the plane
4 x − 7y + 2z = 5