Lesson 4: Lines and Planes (slides + notes)

Lesson 4Lines and Planes

Math 20

September 26, 2007

Announcements

I Problem Set 1 is due today

I Problem Set 2 is on the course web site. Due October 3

I My office hours: Mondays 1–2, Tuesdays 3–4, Wednesdays1–3 (SC 323)

Lines in the plane

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?

Math 20 - September 26, 2007.GWBWednesday, Sep 26, 2007

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Lines in the plane

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.

a

v

Let

a =

(0b

)v =

(1m

)

Then the line can be described as the set of all

x = a + tv =

(0b

)+ t

(1m

)=

(t

mt + b

)as t ranges over all real numbers.



a

v

Let

a =

(0b

)

v =

(1m

)


x = a + tv =

(0b

)+ t

(1m

)=

(t

mt + b




a

v

Let

a =

(0b

)v =

(1m

)


x = a + tv =

(0b

)+ t

(1m

)=

(t

mt + b



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a

v

Let

a =

(0b

)v =

(1m

)


x = a + tv =

(0b

)+ t

(1m

)=

(t

mt + b


Generalizing

Any line in Rn can be described by a point a and a direction v andgiven parametrically by the equation

x = a + tv

Alternatively, any line in Rn can be described by two points a andb by letting a be the point and b− a the direction. Then

x = a + t(b− a) = (1− t)a + tb.


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Applying the definition

Example

Determine if the points a = (1, 2, 3), b = (3, 5, 7), andc = (4, 8, 11) in R3 are on the same line.

SolutionThey are on the same line if c is on the line specified by a and b.So we will find the equation for this line and test if c is on it.The line has a on it and goes in the direction b− a. So it can bewritten in the form

x =

123

+ t

234

=

1 + 2t2 + 3t3 + 4t

Solution (continued)

c is on this line if this system of equations has a solution:

1 + 2t = 5

2 + 3t = 8

3 + 4t = 11

The first one tells us t = 3/2, but the second t = 2. So there is nosolution of all three.


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Generalizing


x = a + tv

Alternatively, any line in Rn can be described by two points a andb by letting a be the point and b− a the direction.

Then

x = a + t(b− a) = (1− t)a + tb.


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Generalizing


x = a + tv

Alternatively, any line in Rn can be described by two points a andb by letting a be the point and b− a the direction. Then

x = a + t(b− a) = (1− t)a + tb.

Lines in the plane, again

a

vp

x

x−a

Let p be perpendicular to v.

Then the head of x is on theline exactly when x− a isparallel to v, or perpendicularto p.

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

p · (x− a) = 0

Lines in the plane, again

a

vp

x

x−a Let p be perpendicular to v.

Then the head of x is on theline exactly when x− a isparallel to v, or perpendicularto p.

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

p · (x− a) = 0

Generalizing again

This generalizes to R3 as well.

x

y

z

a

p

This time, the “locus” is a plane.

Example

Find the equation of the plane that passes through the points(1, 2, 3), (3, 5, 7), and (4, 3, 1)


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Hyperplanes in Rn

DefinitionA hyperplane through a that is orthogonal to a vector p 6= 0 isthe set of all points x satisfying

p · (x− a) = 0.

Lesson 4: Lines and Planes (slides + notes)

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Transcript of Lesson 4: Lines and Planes (slides + notes)