Section 9.1
Coordinates and Distance
Math 21a
February 4, 2008
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Outline
Bingo
Axes and Coordinates in space
Axes
Orientation
Coordinate lines and planes
Distance
The Pythagorean Theorem
Simple curves and surfaces

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

Dimensions

Dimensions

Axes in Flatland
y
x

Axes in Spaceland
z
y
x

Axes in Spaceland
z z
y
y x
x

Axes in Spaceland
z z
y
y x
x
y
x
z

Axes in Spaceland
z z
y
y x
x
x
y
y
x
z
z

Mirror-image axes
z z
y y
x x

Mirror-image axes
z z
y y
x x

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

The right-hand rule
z
x
y

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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).