Section 9.5
Equations of Lines and Planes
Math 21a
February 11, 2008
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Office Hours Tuesday, Wednesday, 2–4pm (SC 323)
All homework on the website
No class Monday 2/18

Outline
Parallel and perpendicular in spaceland
Lines in spaceland
Lines in flatland
Equations for lines in spaceland
Equations for planes
Lines in flatland, again
Planes in spaceland
Distances
Point to line
Point to plane
line to line

parallel and perpendicular quiz
Determine whether each statement is true or false.
1. Two lines parallel to a third line are parallel.
2. Two lines perpendicular to a third line are parallel.
3. Two planes parallel to a third plane are parallel.
4. Two planes perpendicular to a third plane are parallel.
5. Two lines parallel to a plane are parallel.
6. Two lines perpendicular to a plane are parallel.
7. Two planes parallel to a line are parallel.
8. Two planes perpendicular to a line are parallel.
9. Two planes either intersect or are parallel.
10. Two lines either intersect or are parallel.
11. A plane and a line either intersect or are parallel.

parallel and perpendicular quiz
Determine whether each statement is true or false.
1. Two lines parallel to a third line are parallel. true
2. Two lines perpendicular to a third line are parallel.
3. Two planes parallel to a third plane are parallel.
4. Two planes perpendicular to a third plane are parallel.
5. Two lines parallel to a plane are parallel.
6. Two lines perpendicular to a plane are parallel.
7. Two planes parallel to a line are parallel.
8. Two planes perpendicular to a line are parallel.
9. Two planes either intersect or are parallel.
10. Two lines either intersect or are parallel.
11. A plane and a line either intersect or are parallel.

parallel and perpendicular quiz
Determine whether each statement is true or false.
1. Two lines parallel to a third line are parallel. true
2. Two lines perpendicular to a third line are parallel. false
3. Two planes parallel to a third plane are parallel.
4. Two planes perpendicular to a third plane are parallel.
5. Two lines parallel to a plane are parallel.
6. Two lines perpendicular to a plane are parallel.
7. Two planes parallel to a line are parallel.
8. Two planes perpendicular to a line are parallel.
9. Two planes either intersect or are parallel.
10. Two lines either intersect or are parallel.
11. A plane and a line either intersect or are parallel.

parallel and perpendicular quiz
Determine whether each statement is true or false.
1. Two lines parallel to a third line are parallel. true
2. Two lines perpendicular to a third line are parallel. false
3. Two planes parallel to a third plane are parallel. true
4. Two planes perpendicular to a third plane are parallel.
5. Two lines parallel to a plane are parallel.
6. Two lines perpendicular to a plane are parallel.
7. Two planes parallel to a line are parallel.
8. Two planes perpendicular to a line are parallel.
9. Two planes either intersect or are parallel.
10. Two lines either intersect or are parallel.
11. A plane and a line either intersect or are parallel.

parallel and perpendicular quiz
Determine whether each statement is true or false.
1. Two lines parallel to a third line are parallel. true
2. Two lines perpendicular to a third line are parallel. false
3. Two planes parallel to a third plane are parallel. true
4. Two planes perpendicular to a third plane are parallel. false
5. Two lines parallel to a plane are parallel.
6. Two lines perpendicular to a plane are parallel.
7. Two planes parallel to a line are parallel.
8. Two planes perpendicular to a line are parallel.
9. Two planes either intersect or are parallel.
10. Two lines either intersect or are parallel.
11. A plane and a line either intersect or are parallel.

parallel and perpendicular quiz
Determine whether each statement is true or false.
1. Two lines parallel to a third line are parallel. true
2. Two lines perpendicular to a third line are parallel. false
3. Two planes parallel to a third plane are parallel. true
4. Two planes perpendicular to a third plane are parallel. false
5. Two lines parallel to a plane are parallel. false
6. Two lines perpendicular to a plane are parallel.
7. Two planes parallel to a line are parallel.
8. Two planes perpendicular to a line are parallel.
9. Two planes either intersect or are parallel.
10. Two lines either intersect or are parallel.
11. A plane and a line either intersect or are parallel.

parallel and perpendicular quiz
Determine whether each statement is true or false.
1. Two lines parallel to a third line are parallel. true
2. Two lines perpendicular to a third line are parallel. false
3. Two planes parallel to a third plane are parallel. true
4. Two planes perpendicular to a third plane are parallel. false
5. Two lines parallel to a plane are parallel. false
6. Two lines perpendicular to a plane are parallel. true
7. Two planes parallel to a line are parallel.
8. Two planes perpendicular to a line are parallel.
9. Two planes either intersect or are parallel.
10. Two lines either intersect or are parallel.
11. A plane and a line either intersect or are parallel.

parallel and perpendicular quiz
Determine whether each statement is true or false.
1. Two lines parallel to a third line are parallel. true
2. Two lines perpendicular to a third line are parallel. false
3. Two planes parallel to a third plane are parallel. true
4. Two planes perpendicular to a third plane are parallel. false
5. Two lines parallel to a plane are parallel. false
6. Two lines perpendicular to a plane are parallel. true
7. Two planes parallel to a line are parallel. false
8. Two planes perpendicular to a line are parallel.
9. Two planes either intersect or are parallel.
10. Two lines either intersect or are parallel.
11. A plane and a line either intersect or are parallel.

parallel and perpendicular quiz
Determine whether each statement is true or false.
1. Two lines parallel to a third line are parallel. true
2. Two lines perpendicular to a third line are parallel. false
3. Two planes parallel to a third plane are parallel. true
4. Two planes perpendicular to a third plane are parallel. false
5. Two lines parallel to a plane are parallel. false
6. Two lines perpendicular to a plane are parallel. true
7. Two planes parallel to a line are parallel. false
8. Two planes perpendicular to a line are parallel. true
9. Two planes either intersect or are parallel.
10. Two lines either intersect or are parallel.
11. A plane and a line either intersect or are parallel.

parallel and perpendicular quiz
Determine whether each statement is true or false.
1. Two lines parallel to a third line are parallel. true
2. Two lines perpendicular to a third line are parallel. false
3. Two planes parallel to a third plane are parallel. true
4. Two planes perpendicular to a third plane are parallel. false
5. Two lines parallel to a plane are parallel. false
6. Two lines perpendicular to a plane are parallel. true
7. Two planes parallel to a line are parallel. false
8. Two planes perpendicular to a line are parallel. true
9. Two planes either intersect or are parallel. true
10. Two lines either intersect or are parallel.
11. A plane and a line either intersect or are parallel.

parallel and perpendicular quiz
Determine whether each statement is true or false.
1. Two lines parallel to a third line are parallel. true
2. Two lines perpendicular to a third line are parallel. false
3. Two planes parallel to a third plane are parallel. true
4. Two planes perpendicular to a third plane are parallel. false
5. Two lines parallel to a plane are parallel. false
6. Two lines perpendicular to a plane are parallel. true
7. Two planes parallel to a line are parallel. false
8. Two planes perpendicular to a line are parallel. true
9. Two planes either intersect or are parallel. true
10. Two lines either intersect or are parallel. false
11. A plane and a line either intersect or are parallel.

parallel and perpendicular quiz
Determine whether each statement is true or false.
1. Two lines parallel to a third line are parallel. true
2. Two lines perpendicular to a third line are parallel. false
3. Two planes parallel to a third plane are parallel. true
4. Two planes perpendicular to a third plane are parallel. false
5. Two lines parallel to a plane are parallel. false
6. Two lines perpendicular to a plane are parallel. true
7. Two planes parallel to a line are parallel. false
8. Two planes perpendicular to a line are parallel. true
9. Two planes either intersect or are parallel. true
10. Two lines either intersect or are parallel. false
11. A plane and a line either intersect or are parallel. true

Parallelism in spaceland
Two planes are parallel if they do not intersect
A line and a plane are parallel if they do not intersect
Two lines are skew if they are not both contained in a single
plane
Two lines are parallel if they are contained in a common plane
and they do not intersect

Outline
Parallel and perpendicular in spaceland
Lines in spaceland
Lines in flatland
Equations for lines in spaceland
Equations for planes
Lines in flatland, again
Planes in spaceland
Distances
Point to line
Point to plane
line to line

Lines in flatland
There are many ways to specify a line in the plane:

Lines in flatland
There are many ways to specify a line in the plane:
two points
point and slope
slope and intercept

Lines in flatland
There are many ways to specify a line in the plane:
two points
point and slope
slope and intercept
How can we specify a line in three or more dimensions?

Using vectors to describe lines
Let y = mx + b be a line in the plane.
Let

Using vectors to describe lines
Let y = mx + b be a line in the plane.
Let
r0 = 0, b
r0

Using vectors to describe lines
Let y = mx + b be a line in the plane.
Let
v r0 = 0, b v = 1, m
r0

Using vectors to describe lines
Let y = mx + b be a line in the plane.
Let
v r0 = 0, b v = 1, m
r0
Then the line can be described as the set of all
r(t) = r0 + tv
as t ranges over all real numbers.

Lines in spaceland
Any line in R3 can be described by a point with position vector
r0 and a direction vector v. It’s given by the vector equation
r(t) = r0 + tv

Lines in spaceland
Any line in R3 can be described by a point with position vector
r0 and a direction vector v. It’s given by the vector equation
r(t) = r0 + tv
If r = x0 , y0 , z0 and v = a, b, c , then the vector equation
can be rewritten
x, y , z = x0 + ta, y0 + tb, z0 + tc
=⇒ x = x0 + at y = y0 + bt z = z0 + ct
These are called the parametric equations for the line.

Lines in spaceland
Any line in R3 can be described by a point with position vector
r0 and a direction vector v. It’s given by the vector equation
r(t) = r0 + tv
If r = x0 , y0 , z0 and v = a, b, c , then the vector equation
can be rewritten
x, y , z = x0 + ta, y0 + tb, z0 + tc
=⇒ x = x0 + at y = y0 + bt z = z0 + ct
These are called the parametric equations for the line.
Solving the parametric equations for t gives
x − x0 y − y0 z − z0
= =
a b c
These are called the symmetric equations for the line.

Applying the definition
Example
Find the vector, parametric, and symmetric equations for the line
that passes through (1, 2, 3) and (2, 3, 4).

Applying the definition
Example
Find the vector, parametric, and symmetric equations for the line
that passes through (1, 2, 3) and (2, 3, 4).
Solution
Use the initial vector 1, 2, 3 and direction vector
2, 3, 4 − 1, 2, 3 = 1, 1, 1 . Hence
r(t) = 1, 2, 3 + t 1, 1, 1
The parametric equations are
x =1+t y =2+t z =3+t
The symmetric equations are
x −1=y −2=z −3

Another vector equation
Alternatively, any line in R3 can be described by two points with
position vectors r0 and r1 by letting r0 be the point and r1 − r0 the
direction.

Another vector equation
Alternatively, any line in R3 can be described by two points with
position vectors r0 and r1 by letting r0 be the point and r1 − r0 the
direction. Then
x = r0 + t(r1 − r0 ) = (1 − t)r0 + tr1 .

Outline
Parallel and perpendicular in spaceland
Lines in spaceland
Lines in flatland
Equations for lines in spaceland
Equations for planes
Lines in flatland, again
Planes in spaceland
Distances
Point to line
Point to plane
line to line

Lines in flatland, again
Let n be perpendicular to v.
n v
r0

Lines in flatland, again
r0 Let n be perpendicular to v.
r− Then the head of r is on the
n v line exactly when r − r0 is
r parallel to v, or perpendicular
r0
to n.

Lines in flatland, again
r0 Let n be perpendicular to v.
r− Then the head of r is on the
n v line exactly when r − r0 is
r parallel to v, or perpendicular
r0
to n.
So the line can be described as the set of all r such that
n · (r − r0 ) = 0

Generalizing again
This generalizes to spaceland as well.
z
n
r0
y
x

Generalizing again
This generalizes to spaceland as well.
z
n
r0
y
x

Generalizing again
This generalizes to spaceland as well.
z
n
r0
y
x
This time, the locus is a plane.

Equations for planes
The plane passing through the point with position vector
r0 = x0 , y0 , z0 perpendicular to a, b, c has equations:
The vector equation
n · (r − r0 ) = 0 ⇐⇒ n · r = n · r0

Equations for planes
The plane passing through the point with position vector
r0 = x0 , y0 , z0 perpendicular to a, b, c has equations:
The vector equation
n · (r − r0 ) = 0 ⇐⇒ n · r = n · r0
Rewriting the dot product in component terms gives the
scalar equation
a(x − x0 ) + b(y − y0 ) + c(z − z0 ) = 0
The vector n is called a normal vector to the plane.

Equations for planes
The plane passing through the point with position vector
r0 = x0 , y0 , z0 perpendicular to a, b, c has equations:
The vector equation
n · (r − r0 ) = 0 ⇐⇒ n · r = n · r0
Rewriting the dot product in component terms gives the
scalar equation
a(x − x0 ) + b(y − y0 ) + c(z − z0 ) = 0
The vector n is called a normal vector to the plane.
Rearranging this gives the linear equation
ax + by + cz + d = 0,
where d = −ax0 − by0 − cz0 .

Example
Find an equation of the plane that passes through the points
P(1, 2, 3), Q(3, 5, 7), and R(4, 3, 1).

Example
Find an equation of the plane that passes through the points
P(1, 2, 3), Q(3, 5, 7), and R(4, 3, 1).
Solution −→ −→ − →
Let r0 = OP = 1, 2, 3 . To get n, take PQ × PR:
i j k
−→ − →
PQ × PR = 2 3 4 = −10, 16, −7
3 1 −2
So the scalar equation is
−10(x − 1) + 16(y − 2) − 7(z − 3) = 0.

Outline
Parallel and perpendicular in spaceland
Lines in spaceland
Lines in flatland
Equations for lines in spaceland
Equations for planes
Lines in flatland, again
Planes in spaceland
Distances
Point to line
Point to plane
line to line

Distance from point to line
Definition
The distance between a point and a line is the smallest distance
from that point to all points on the line. You can find it by
projection.
Q
v
θ
P0

Distance from point to line
Definition
The distance between a point and a line is the smallest distance
from that point to all points on the line. You can find it by
projection.
Q
b
v
θ
P0

Distance from point to line
Definition
The distance between a point and a line is the smallest distance
from that point to all points on the line. You can find it by
projection.
Q
b
b·v
projv b = v
v v·v
θ
P0

Distance from point to line
Definition
The distance between a point and a line is the smallest distance
from that point to all points on the line. You can find it by
projection.
Q
b·v
b− v
v·v
b
b·v
projv b = v
v v·v
θ
P0

Example
Find the distance between the point (4, 6) and the line
x − 2y + 3 = 0.

Example
Find the distance between the point (4, 6) and the line
x − 2y + 3 = 0.
Solution
The line goes through (1, 2) and has slope 1/2, so we can use
v = 2, 1 and b = 3, 4 . Then the projection of b on the line is
given by
b·v 10
projv b = v= 2, 1 = 4, 2
v·v 5
So
b − projv b = 3, 4 − 4, 2 = −1, 2
(Notice that 2, 1 and −1, 2 are perpendicular.) So the distance
is √
| −1, 2 | = 5

Point to plane
Definition
The distance between a point and a plane is the smallest distance
from that point to all points on the line.
Q
b |n · b|
|n|
n
P0
To find the distance from the a point to a plane, project the
displacement vector from any point on the plane to the given point
onto the normal vector.

We have
|n · b|
D=
|n|
If Q = (x1 , y1 , z1 ), and the plane is given by ax + by + cz + d = 0,
then n = a, b, c , and
n · b = a, b, c · x1 − x0 , y1 − y0 , z1 − z0
= ax1 + by1 + cz1 − ax0 − by0 − cz0
= ax1 + by1 + cz1 + d

We have
|n · b|
D=
|n|
If Q = (x1 , y1 , z1 ), and the plane is given by ax + by + cz + d = 0,
then n = a, b, c , and
n · b = a, b, c · x1 − x0 , y1 − y0 , z1 − z0
= ax1 + by1 + cz1 − ax0 − by0 − cz0
= ax1 + by1 + cz1 + d
So the distance between the plane ax + by + cz + d = 0 and the
point (x1 , y1 , z1 ) is
|ax1 + by1 + cz1 + d|
D= √
a2 + b 2 + c 2

Example
Find the distance between the plane containing the three points
P(1, 2, 3), Q(3, 5, 7), and R(4, 3, 1) and the origin.

Example
Find the distance between the plane containing the three points
P(1, 2, 3), Q(3, 5, 7), and R(4, 3, 1) and the origin.
Solution
We’ve already found the plane has scalar equation given by
0 = −10(x − 1) + 16(y − 2) − 7(z − 3)
= −10x + 16y − 7z − 1
So d = 1. Using the formula above with (x1 , y1 , z1 ) = (0, 0, 0) we
have
1 1
D=√ = √
102 + 162 + 72 9 5

line to line
To find the distance between two skew lines, create two parallel
planes and find the distance between a point in one to the other.
For an example, see Example 10 on page 673.