The document summarizes key concepts about coordinates and distance in flatland and spaceland: - It introduces coordinate axes and how to place points in flatland and spaceland using coordinates. - It covers the Pythagorean theorem and how to calculate distance between two points in flatland and spaceland. - It describes how loci of points at a fixed distance from a fixed point form circles in flatland and spheres in spaceland. - It provides an example of using an equation to represent a spherical surface in space.

1. Section 9.1
Coordinates and Distance
Math 21a
February 4, 2008
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Homework for Wednesday 2/6:
9.1: 5, 6, 7, 8, 10, 14, 18, 30, 32, 34;
9.2.1*, 9.2.3*, 9.3.1*

2. Outline
Bingo
Axes and Coordinates in space
Axes
Orientation
Coordinate lines and planes
Distance
The Pythagorean Theorem
Simple curves and surfaces

3. Outline
Bingo
Axes and Coordinates in space
Axes
Orientation
Coordinate lines and planes
Distance
The Pythagorean Theorem
Simple curves and surfaces

4. Dimensions

5. Dimensions

6. Axes in Flatland
y
x

7. Axes in Spaceland
z
y
x

8. Axes in Spaceland
z z
y
y x
x

9. Axes in Spaceland
z z
y
y x
x
y
x
z

10. Axes in Spaceland
z z
y
y x
x
x
y
y
x
z
z

11. Mirror-image axes
z z
y y
x x

12. Mirror-image axes
z z
y y
x x

13. Mirror-image axes
z z
y y
x x
Our convention is only to choose axes like those on the right.

14. The right-hand rule
z
x
y

15. Placing points—Flatland
Example
Place the point P(3, 4) in the plane.

16. Placing points—Flatland
Example
Place the point P(3, 4) in the plane.
Solution
y
x

17. Placing points—Flatland
Example
Place the point P(3, 4) in the plane.
Solution
y
x
3

18. Placing points—Flatland
Example
Place the point P(3, 4) in the plane.
Solution
y
4
x
3

19. Placing points—Flatland
Example
Place the point P(3, 4) in the plane.
Solution
y
4
x
3

20. Placing points—spaceland
Example
Place the point P(3, 4, 5) in space.

21. Placing points—spaceland
Example
Place the point P(3, 4, 5) in space.
Solution
z
y
x

22. Placing points—spaceland
Example
Place the point P(3, 4, 5) in space.
Solution
z
y
|
|
3x
|

23. Placing points—spaceland
Example
Place the point P(3, 4, 5) in space.
Solution
z
| | | | y
4
|
|
3x
|

24. Placing points—spaceland
Example
Place the point P(3, 4, 5) in space.
Solution
−5
z
−
−
−
−
| | | | y
4
|
|
3x
|

25. Meet the Mathematician: Ren´ Descartes
e
French, 1596–1650
Philosopher and
mathematician
Cogito ergo sum
Cartesian coordinate
system

26. Coordinate lines in flatland
Example
Draw the line x = 3.

27. Coordinate lines in flatland
Example
Draw the line x = 3.
Solution
y
x

28. Coordinate lines in flatland
Example
Draw the line x = 3.
Solution
y
x
(3, 0)

29. Coordinate planes in spaceland
Example
Draw the plane x = 3.

30. Coordinate planes in spaceland
Example
Draw the plane x = 3.
Solution
z
y
x

31. Coordinate planes in spaceland
Example
Draw the plane x = 3.
Solution
z
y
x
(3, 0, 0)

32. Outline
Bingo
Axes and Coordinates in space
Axes
Orientation
Coordinate lines and planes
Distance
The Pythagorean Theorem
Simple curves and surfaces

33. The Pythagorean Theorem
If a, b, and c are sides of a
right triangle and c is the
hypotenuse, then
a2 + b 2 = c 2

34. Meet the mathematician: Pythagoras
Greek, c. 580 – c. 490
BCE (pre-Socratic)
Philosopher who believed
all order is in number
until one of his order
discovered irrational
numbers

35. Meet the mathematician: Pythagoras
Greek, c. 580 – c. 490
BCE (pre-Socratic)
Philosopher who believed
all order is in number
until one of his order
discovered irrational
numbers

36. Distance in flatland
Given two points P1 (x1 , y1 ) and P2 (x2 , y2 ), we can use Pythagoras
to find the distance between them:
P2
y
y2 − y1
P1 x2 − x1
x
|P1 P2 | = (x2 − x1 )2 + (y2 − y1 )2

37. Distance in spaceland
Example
Find the distance between the points P1 (3, 2, 1) and P2 (4, 4, 4).

38. Distance in spaceland
Example
Find the distance between the points P1 (3, 2, 1) and P2 (4, 4, 4).
Solution
z
P2
y
P1
x

39. Distance in spaceland
Example
Find the distance between the points P1 (3, 2, 1) and P2 (4, 4, 4).
Solution
z
P2
y
2
1
P1
x

40. Distance in spaceland
Example
Find the distance between the points P1 (3, 2, 1) and P2 (4, 4, 4).
Solution
z
P2
y
√2
1 5
P1
x

41. Distance in spaceland
Example
Find the distance between the points P1 (3, 2, 1) and P2 (4, 4, 4).
Solution
z
P2
d 3
y
√2
1 5
P1
x

42. Distance in spaceland—General
Theorem
The distance between (x1 , y1 , z1 ) and (x2 , y2 , z2 ) is
(x2 − x1 )2 + (y2 − y1 )2 + (z2 − z1 )2

43. A curve
In flatland, the set (or locus) of all points which are a fixed
distance from a fixed point is a

44. A curve
In flatland, the set (or locus) of all points which are a fixed
distance from a fixed point is a circle.
y
x

45. A surface
In spaceland, the locus of all points which are a fixed distance from
a fixed point is a

46. A surface
In spaceland, the locus of all points which are a fixed distance from
a fixed point is a sphere.

47. A surface
In spaceland, the locus of all points which are a fixed distance from
a fixed point is a sphere.

48. Munging an equation to see its surface
Example
Find the surface is represented by the equation
x 2 + y 2 + z 2 − 4x + 8y − 10z + 36 = 0

49. Munging an equation to see its surface
Example
Find the surface is represented by the equation
x 2 + y 2 + z 2 − 4x + 8y − 10z + 36 = 0
Solution
We can complete the square:
0 = x 2 − 4x + 4 + y 2 + 8y + 16 + z 2 − 10z + 25 + 36 − 4 − 16 − 25
= (x − 2)2 + (y + 4)2 + (z − 5)2 − 9

50. Munging an equation to see its surface
Example
Find the surface is represented by the equation
x 2 + y 2 + z 2 − 4x + 8y − 10z + 36 = 0
Solution
We can complete the square:
0 = x 2 − 4x + 4 + y 2 + 8y + 16 + z 2 − 10z + 25 + 36 − 4 − 16 − 25
= (x − 2)2 + (y + 4)2 + (z − 5)2 − 9
So
(x − 2)2 + (y + 4) + (z − 5)2 = 9
=⇒ |(x, y , z)(2, −4, 5)| = 3

51. Munging an equation to see its surface
Example
Find the surface is represented by the equation
x 2 + y 2 + z 2 − 4x + 8y − 10z + 36 = 0
Solution
We can complete the square:
0 = x 2 − 4x + 4 + y 2 + 8y + 16 + z 2 − 10z + 25 + 36 − 4 − 16 − 25
= (x − 2)2 + (y + 4)2 + (z − 5)2 − 9
So
(x − 2)2 + (y + 4) + (z − 5)2 = 9
=⇒ |(x, y , z)(2, −4, 5)| = 3
This is a sphere of radius 3, centered at (2, −4, 5).