The document discusses lines and planes in mathematics. It provides multiple ways to specify a line, including using two points, a point and slope, or a slope and y-intercept. Lines can also be described using vectors, with a line being the set of points a + tv, where a is a point on the line, v is a direction vector, and t is a real number. Planes are similarly defined as the set of points where the dot product of a normal vector p and the offset (x - a) is 0, where a is a point on the plane. An example shows how to check if three points lie on the same line by finding the line equation and checking if a third point satisfies it.

1. Lesson 4
Lines and Planes
Math 20
September 26, 2007
Announcements
Problem Set 1 is due today
Problem Set 2 is on the course web site. Due October 3
My office hours: Mondays 1–2, Tuesdays 3–4, Wednesdays
1–3 (SC 323)

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

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

5. Lines in the plane
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?

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

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

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

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

11. Generalizing
Any line in Rn can be described by a point a and a direction v and
given parametrically by the equation
x = a + tv

13. Applying the definition
Example
Determine if the points a = (1, 2, 3), b = (3, 5, 7), and
c = (4, 8, 11) in R3 are on the same line.

14. Applying the definition
Example
Determine if the points a = (1, 2, 3), b = (3, 5, 7), and
c = (4, 8, 11) in R3 are on the same line.
Solution
They 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 be
written in the form
     
1 2 1 + 2t
x = 2 + t 3 = 2 + 3t 
3 4 3 + 4t

15. 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 no
solution of all three.

18. Generalizing
Any line in Rn can be described by a point a and a direction v and
given parametrically by the equation
x = a + tv
Alternatively, any line in Rn can be described by two points a and
b by letting a be the point and b − a the direction.

20. Generalizing
Any line in Rn can be described by a point a and a direction v and
given parametrically by the equation
x = a + tv
Alternatively, any line in Rn can be described by two points a and
b by letting a be the point and b − a the direction. Then
x = a + t(b − a) = (1 − t)a + tb.

21. Lines in the plane, again
Let p be perpendicular to v.
p v
a

22. Lines in the plane, again
a Let p be perpendicular to v.
x− Then the head of x is on the
p v line exactly when x − a is
x parallel to v, or perpendicular
a to p.

23. Lines in the plane, again
a Let p be perpendicular to v.
x− Then the head of x is on the
p v line exactly when x − a is
x parallel to v, or perpendicular
a to p.
So the line can be described as the set of all x such that
p · (x − a) = 0

24. Generalizing again
This generalizes to R3 as well.
z
p
a
y
x

25. Generalizing again
This generalizes to R3 as well.
z
p
a
y
x

26. Generalizing again
This generalizes to R3 as well.
z
p
a
y
x
This time, the “locus” is a plane.

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

31. Hyperplanes in Rn
Definition
A hyperplane through a that is orthogonal to a vector p = 0 is
the set of all points x satisfying
p · (x − a) = 0.