Lesson 4: Lines, Planes, and the Distance Formula

Section 9.5Equations of Lines and Planes

Math 21a

February 11, 2008

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I All homework on the website

I No class Monday 2/18

Outline

Parallel and perpendicular in spaceland

Lines in spacelandLines in flatlandEquations for lines in spaceland

Equations for planesLines in flatland, againPlanes in spaceland

DistancesPoint to linePoint to planeline to line

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

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

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

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

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

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

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

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

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

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

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

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

I Two planes are parallel if they do not intersect

I A line and a plane are parallel if they do not intersect

I Two lines are skew if they are not both contained in a singleplane

I Two lines are parallel if they are contained in a common planeand they do not intersect

Outline

Parallel and perpendicular in spaceland

Lines in spacelandLines in flatlandEquations for lines in spaceland

Equations for planesLines in flatland, againPlanes in spaceland

DistancesPoint to linePoint to planeline to line

Lines in flatland

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

I two points

I point and slope

I slope and intercept

How can we specify a line in three or more dimensions?

Lines in flatland

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

I two points

I point and slope

I slope and intercept

How can we specify a line in three or more dimensions?

Lines in flatland

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

I two points

I point and slope

I 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.

r0

v

Let

r0 = 〈0, b〉 v = 〈1,m〉

Then the line can be described as the set of all

r(t) = r0 + tv

as t ranges over all real numbers.

Using vectors to describe lines

Let y = mx + b be a line in the plane.

r0

v

Let

r0 = 〈0, b〉

v = 〈1,m〉

Then the line can be described as the set of all

r(t) = r0 + tv

as t ranges over all real numbers.

Using vectors to describe lines

Let y = mx + b be a line in the plane.

r0

v

Let

r0 = 〈0, b〉 v = 〈1,m〉

Then the line can be described as the set of all

r(t) = r0 + tv

as t ranges over all real numbers.

Using vectors to describe lines

Let y = mx + b be a line in the plane.

r0

v

Let

r0 = 〈0, b〉 v = 〈1,m〉

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

I Any line in R3 can be described by a point with position vectorr0 and a direction vector v. It’s given by the vector equation

r(t) = r0 + tv

I If r = 〈x0, y0, z0〉 and v = 〈a, b, c〉, then the vector equationcan 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.

I Solving the parametric equations for t gives

x − x0

a=

y − y0

b=

z − z0

c

These are called the symmetric equations for the line.

Lines in spaceland

I Any line in R3 can be described by a point with position vectorr0 and a direction vector v. It’s given by the vector equation

r(t) = r0 + tv

I If r = 〈x0, y0, z0〉 and v = 〈a, b, c〉, then the vector equationcan 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.

I Solving the parametric equations for t gives

x − x0

a=

y − y0

b=

z − z0

c

These are called the symmetric equations for the line.

Lines in spaceland

I Any line in R3 can be described by a point with position vectorr0 and a direction vector v. It’s given by the vector equation

r(t) = r0 + tv

I If r = 〈x0, y0, z0〉 and v = 〈a, b, c〉, then the vector equationcan 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.

I Solving the parametric equations for t gives

x − x0

a=

y − y0

b=

z − z0

c

These are called the symmetric equations for the line.

Applying the definition

Example

Find the vector, parametric, and symmetric equations for the linethat passes through (1, 2, 3) and (2, 3, 4).

Solution

I 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〉

I The parametric equations are

x = 1 + t y = 2 + t z = 3 + t

I The symmetric equations are

x − 1 = y − 2 = z − 3

Applying the definition

Example

Find the vector, parametric, and symmetric equations for the linethat passes through (1, 2, 3) and (2, 3, 4).

Solution

I 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〉

I The parametric equations are

x = 1 + t y = 2 + t z = 3 + t

I The symmetric equations are

x − 1 = y − 2 = z − 3

Another vector equation

Alternatively, any line in R3 can be described by two points withposition vectors r0 and r1 by letting r0 be the point and r1 − r0 thedirection.

Then

x = r0 + t(r1 − r0) = (1− t)r0 + tr1.

Another vector equation

Alternatively, any line in R3 can be described by two points withposition vectors r0 and r1 by letting r0 be the point and r1 − r0 thedirection. Then

x = r0 + t(r1 − r0) = (1− t)r0 + tr1.

Outline

Parallel and perpendicular in spaceland

Lines in spacelandLines in flatlandEquations for lines in spaceland

Equations for planesLines in flatland, againPlanes in spaceland

DistancesPoint to linePoint to planeline to line

Lines in flatland, again

r0

vn

r

r −r0

Let n be perpendicular to v.

Then the head of r is on theline exactly when r − r0 isparallel to v, or perpendicularto n.

So the line can be described as the set of all r such that

n · (r − r0) = 0

Lines in flatland, again

r0

vn

r

r −r0 Let n be perpendicular to v.

Then the head of r is on theline exactly when r − r0 isparallel to v, or perpendicularto n.

So the line can be described as the set of all r such that

n · (r − r0) = 0

Lines in flatland, again

r0

vn

r

r −r0 Let n be perpendicular to v.

Then the head of r is on theline exactly when r − r0 isparallel to v, or perpendicularto 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.

x

y

z

r0

n

This time, the locus is a plane.

Generalizing again

This generalizes to spaceland as well.

x

y

z

r0

n

This time, the locus is a plane.

Generalizing again

This generalizes to spaceland as well.

x

y

z

r0

n

This time, the locus is a plane.

Equations for planes

The plane passing through the point with position vectorr0 = 〈x0, y0, z0〉 perpendicular to 〈a, b, c〉 has equations:

I The vector equation

n · (r − r0) = 0 ⇐⇒ n · r = n · r0

I Rewriting the dot product in component terms gives thescalar equation

a(x − x0) + b(y − y0) + c(z − z0) = 0

The vector n is called a normal vector to the plane.

I Rearranging this gives the linear equation

ax + by + cz + d = 0,

where d = −ax0 − by0 − cz0.

Equations for planes

The plane passing through the point with position vectorr0 = 〈x0, y0, z0〉 perpendicular to 〈a, b, c〉 has equations:

I The vector equation

n · (r − r0) = 0 ⇐⇒ n · r = n · r0

I Rewriting the dot product in component terms gives thescalar equation

a(x − x0) + b(y − y0) + c(z − z0) = 0

The vector n is called a normal vector to the plane.

I Rearranging this gives the linear equation

ax + by + cz + d = 0,

where d = −ax0 − by0 − cz0.

Equations for planes

The plane passing through the point with position vectorr0 = 〈x0, y0, z0〉 perpendicular to 〈a, b, c〉 has equations:

I The vector equation

n · (r − r0) = 0 ⇐⇒ n · r = n · r0

I Rewriting the dot product in component terms gives thescalar equation

a(x − x0) + b(y − y0) + c(z − z0) = 0

The vector n is called a normal vector to the plane.

I 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 pointsP(1, 2, 3), Q(3, 5, 7), and R(4, 3, 1).

SolutionLet r0 =

−→OP = 〈1, 2, 3〉. To get n, take

−→PQ ×

−→PR:

−→PQ ×

−→PR =

∣∣∣∣∣∣i j k2 3 43 1 −2

∣∣∣∣∣∣ = 〈−10, 16,−7〉

So the scalar equation is

−10(x − 1) + 16(y − 2)− 7(z − 3) = 0.

Example

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

SolutionLet r0 =

−→OP = 〈1, 2, 3〉. To get n, take

−→PQ ×

−→PR:

−→PQ ×

−→PR =

∣∣∣∣∣∣i j k2 3 43 1 −2

∣∣∣∣∣∣ = 〈−10, 16,−7〉

So the scalar equation is

−10(x − 1) + 16(y − 2)− 7(z − 3) = 0.

Outline

Parallel and perpendicular in spaceland

Lines in spacelandLines in flatlandEquations for lines in spaceland

Equations for planesLines in flatland, againPlanes in spaceland

DistancesPoint to linePoint to planeline to line

Distance from point to line

DefinitionThe distance between a point and a line is the smallest distancefrom that point to all points on the line. You can find it byprojection.

P0

Q

v

b

θ

projv b =b · vv · v

v

∣∣∣∣b− b · vv · v

v

∣∣∣∣

Distance from point to line

DefinitionThe distance between a point and a line is the smallest distancefrom that point to all points on the line. You can find it byprojection.

P0

Q

v

b

θ

projv b =b · vv · v

v

∣∣∣∣b− b · vv · v

v

∣∣∣∣

Distance from point to line

DefinitionThe distance between a point and a line is the smallest distancefrom that point to all points on the line. You can find it byprojection.

P0

Q

v

b

θ

projv b =b · vv · v

v

∣∣∣∣b− b · vv · v

v

∣∣∣∣

Distance from point to line

DefinitionThe distance between a point and a line is the smallest distancefrom that point to all points on the line. You can find it byprojection.

P0

Q

v

b

θ

projv b =b · vv · v

v

∣∣∣∣b− b · vv · v

v

∣∣∣∣

Example

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

SolutionThe line goes through (1, 2) and has slope 1/2, so we can usev = 〈2, 1〉 and b = 〈3, 4〉. Then the projection of b on the line isgiven by

projv b =b · vv · v

v =10

5〈2, 1〉 = 〈4, 2〉

Sob− projv b = 〈3, 4〉 − 〈4, 2〉 = 〈−1, 2〉

(Notice that 〈2, 1〉 and 〈−1, 2〉 are perpendicular.) So the distanceis

|〈−1, 2〉| =√

5

Example

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

SolutionThe line goes through (1, 2) and has slope 1/2, so we can usev = 〈2, 1〉 and b = 〈3, 4〉. Then the projection of b on the line isgiven by

projv b =b · vv · v

v =10

5〈2, 1〉 = 〈4, 2〉

Sob− projv b = 〈3, 4〉 − 〈4, 2〉 = 〈−1, 2〉

(Notice that 〈2, 1〉 and 〈−1, 2〉 are perpendicular.) So the distanceis

|〈−1, 2〉| =√

5

Point to plane

DefinitionThe distance between a point and a plane is the smallest distancefrom that point to all points on the line.

P0

Q

n

b |n · b||n|

To find the distance from the a point to a plane, project thedisplacement vector from any point on the plane to the given pointonto the normal vector.

We have

D =|n · b||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 thepoint (x1, y1, z1) is

D =|ax1 + by1 + cz1 + d |√

a2 + b2 + c2

We have

D =|n · b||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 thepoint (x1, y1, z1) is

D =|ax1 + by1 + cz1 + d |√

a2 + b2 + c2

Example

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

SolutionWe’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) wehave

D =1√

102 + 162 + 72=

1

9√

5

Example

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

SolutionWe’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) wehave

D =1√

102 + 162 + 72=

1

9√

5

line to line

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