lesson 4: lines and planes (slides + notes)
TRANSCRIPT
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?
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.
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.
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.
Math 20 - September 26, 2007.GWBWednesday, Sep 26, 2007
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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.
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.
Math 20 - September 26, 2007.GWBWednesday, Sep 26, 2007
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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
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.
Math 20 - September 26, 2007.GWBWednesday, Sep 26, 2007
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Math 20 - September 26, 2007.GWBWednesday, Sep 26, 2007
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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.
Math 20 - September 26, 2007.GWBWednesday, Sep 26, 2007
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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.
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
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.
Generalizing again
This generalizes to R3 as well.
x
y
z
a
p
This time, the “locus” is a plane.
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)
Math 20 - September 26, 2007.GWBWednesday, Sep 26, 2007
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Math 20 - September 26, 2007.GWBWednesday, Sep 26, 2007
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Math 20 - September 26, 2007.GWBWednesday, Sep 26, 2007
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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.