Math 141
Lecture 1
Catalin Zara
with modifications by Todor Milev
University of Massachusetts Boston
2014
Math 141
Lecture 1
2014
Outline
1
Space
Math 141
Lecture 1
2014
Space
Configurations of pair of lines
2 lines
pair of objects
intersecting lines
parallel lines
skew lines
Math 141
intersection
?
Lecture 1
parallelism
co-planar?
2014
Space
Configurations of pair of lines
2 lines
pair of objects
intersecting lines
parallel lines
skew lines
Math 141
intersection
one point
Lecture 1
parallelism
co-planar?
2014
Space
Configurations of pair of lines
2 lines
pair of objects
intersecting lines
parallel lines
skew lines
Math 141
intersection
one point
Lecture 1
parallelism
?
co-planar?
2014
Space
Configurations of pair of lines
2 lines
pair of objects
intersecting lines
parallel lines
skew lines
Math 141
intersection
one point
Lecture 1
parallelism
not parallel
co-planar?
2014
Space
Configurations of pair of lines
2 lines
pair of objects
intersecting lines
parallel lines
skew lines
Math 141
intersection
one point
Lecture 1
parallelism
not parallel
co-planar?
?
2014
Space
Configurations of pair of lines
2 lines
pair of objects
intersecting lines
parallel lines
skew lines
Math 141
intersection
one point
Lecture 1
parallelism
not parallel
co-planar?
yes
2014
Space
Configurations of pair of lines
2 lines
pair of objects
intersecting lines
parallel lines
skew lines
Math 141
intersection
one point
?
Lecture 1
parallelism
not parallel
co-planar?
yes
2014
Space
Configurations of pair of lines
2 lines
pair of objects
intersecting lines
parallel lines
skew lines
Math 141
intersection
one point
empty
Lecture 1
parallelism
not parallel
co-planar?
yes
2014
Space
Configurations of pair of lines
2 lines
pair of objects
intersecting lines
parallel lines
skew lines
Math 141
intersection
one point
empty
Lecture 1
parallelism
not parallel
?
co-planar?
yes
2014
Space
Configurations of pair of lines
2 lines
pair of objects
intersecting lines
parallel lines
skew lines
Math 141
intersection
one point
empty
Lecture 1
parallelism
not parallel
parallel
co-planar?
yes
2014
Space
Configurations of pair of lines
2 lines
pair of objects
intersecting lines
parallel lines
skew lines
Math 141
intersection
one point
empty
Lecture 1
parallelism
not parallel
parallel
co-planar?
yes
?
2014
Space
Configurations of pair of lines
2 lines
pair of objects
intersecting lines
parallel lines
skew lines
Math 141
intersection
one point
empty
Lecture 1
parallelism
not parallel
parallel
co-planar?
yes
yes
2014
Space
Configurations of pair of lines
2 lines
pair of objects
intersecting lines
parallel lines
skew lines
Math 141
intersection
one point
empty
?
Lecture 1
parallelism
not parallel
parallel
co-planar?
yes
yes
2014
Space
Configurations of pair of lines
2 lines
pair of objects
intersecting lines
parallel lines
skew lines
Math 141
intersection
one point
empty
none
Lecture 1
parallelism
not parallel
parallel
co-planar?
yes
yes
2014
Space
Configurations of pair of lines
2 lines
pair of objects
intersecting lines
parallel lines
skew lines
Math 141
intersection
one point
empty
none
Lecture 1
parallelism
not parallel
parallel
?
co-planar?
yes
yes
2014
Space
Configurations of pair of lines
2 lines
pair of objects
intersecting lines
parallel lines
skew lines
Math 141
intersection
one point
empty
none
Lecture 1
parallelism
not parallel
parallel
not parallel
co-planar?
yes
yes
2014
Space
Configurations of pair of lines
2 lines
pair of objects
intersecting lines
parallel lines
skew lines
Math 141
intersection
one point
empty
none
Lecture 1
parallelism
not parallel
parallel
not parallel
co-planar?
yes
yes
?
2014
Space
Configurations of pair of lines
2 lines
pair of objects
intersecting lines
parallel lines
skew lines
Math 141
intersection
one point
empty
none
Lecture 1
parallelism
not parallel
parallel
not parallel
co-planar?
yes
yes
no
2014
Space
line &
plane
2 lines
Configurations of pair of lines
pair of objects
intersecting lines
parallel lines
skew lines
line intersecting plane
line parallel to a plane
line lying in plane
Math 141
intersection
one point
empty
none
?
parallelism
not parallel
parallel
not parallel
co-planar?
yes
yes
no
-
Lecture 1
2014
Space
line &
plane
2 lines
Configurations of pair of lines
pair of objects
intersecting lines
parallel lines
skew lines
line intersecting plane
line parallel to a plane
line lying in plane
Math 141
intersection
one point
empty
none
one point
parallelism
not parallel
parallel
not parallel
co-planar?
yes
yes
no
-
Lecture 1
2014
Space
line &
plane
2 lines
Configurations of pair of lines
pair of objects
intersecting lines
parallel lines
skew lines
line intersecting plane
line parallel to a plane
line lying in plane
Math 141
intersection
one point
empty
none
one point
parallelism
not parallel
parallel
not parallel
?
co-planar?
yes
yes
no
-
Lecture 1
2014
Space
line &
plane
2 lines
Configurations of pair of lines
pair of objects
intersecting lines
parallel lines
skew lines
line intersecting plane
line parallel to a plane
line lying in plane
Math 141
intersection
one point
empty
none
one point
parallelism
not parallel
parallel
not parallel
not parallel
co-planar?
yes
yes
no
-
Lecture 1
2014
Space
line &
plane
2 lines
Configurations of pair of lines
pair of objects
intersecting lines
parallel lines
skew lines
line intersecting plane
line parallel to a plane
line lying in plane
Math 141
intersection
one point
empty
none
one point
parallelism
not parallel
parallel
not parallel
not parallel
co-planar?
yes
yes
no
?
-
Lecture 1
2014
Space
line &
plane
2 lines
Configurations of pair of lines
pair of objects
intersecting lines
parallel lines
skew lines
line intersecting plane
line parallel to a plane
line lying in plane
Math 141
intersection
one point
empty
none
one point
parallelism
not parallel
parallel
not parallel
not parallel
co-planar?
yes
yes
no
no
-
Lecture 1
2014
Space
line &
plane
2 lines
Configurations of pair of lines
pair of objects
intersecting lines
parallel lines
skew lines
line intersecting plane
line parallel to a plane
line lying in plane
Math 141
intersection
one point
empty
none
one point
?
parallelism
not parallel
parallel
not parallel
not parallel
co-planar?
yes
yes
no
no
-
Lecture 1
2014
Space
line &
plane
2 lines
Configurations of pair of lines
pair of objects
intersecting lines
parallel lines
skew lines
line intersecting plane
line parallel to a plane
line lying in plane
Math 141
intersection
one point
empty
none
one point
none
parallelism
not parallel
parallel
not parallel
not parallel
co-planar?
yes
yes
no
no
-
Lecture 1
2014
Space
line &
plane
2 lines
Configurations of pair of lines
pair of objects
intersecting lines
parallel lines
skew lines
line intersecting plane
line parallel to a plane
line lying in plane
Math 141
intersection
one point
empty
none
one point
none
Lecture 1
parallelism
not parallel
parallel
not parallel
not parallel
?
-
co-planar?
yes
yes
no
no
2014
Space
line &
plane
2 lines
Configurations of pair of lines
pair of objects
intersecting lines
parallel lines
skew lines
line intersecting plane
line parallel to a plane
line lying in plane
Math 141
intersection
one point
empty
none
one point
none
Lecture 1
parallelism
not parallel
parallel
not parallel
not parallel
parallel
-
co-planar?
yes
yes
no
no
2014
Space
line &
plane
2 lines
Configurations of pair of lines
pair of objects
intersecting lines
parallel lines
skew lines
line intersecting plane
line parallel to a plane
line lying in plane
Math 141
intersection
one point
empty
none
one point
none
Lecture 1
parallelism
not parallel
parallel
not parallel
not parallel
parallel
-
co-planar?
yes
yes
no
no
?
2014
Space
line &
plane
2 lines
Configurations of pair of lines
pair of objects
intersecting lines
parallel lines
skew lines
line intersecting plane
line parallel to a plane
line lying in plane
Math 141
intersection
one point
empty
none
one point
none
Lecture 1
parallelism
not parallel
parallel
not parallel
not parallel
parallel
-
co-planar?
yes
yes
no
no
no
2014
Space
line &
plane
2 lines
Configurations of pair of lines
pair of objects
intersecting lines
parallel lines
skew lines
line intersecting plane
line parallel to a plane
line lying in plane
Math 141
intersection
one point
empty
none
one point
none
?
Lecture 1
parallelism
not parallel
parallel
not parallel
not parallel
parallel
-
co-planar?
yes
yes
no
no
no
2014
Space
line &
plane
2 lines
Configurations of pair of lines
pair of objects
intersecting lines
parallel lines
skew lines
line intersecting plane
line parallel to a plane
line lying in plane
Math 141
intersection
one point
empty
none
one point
none
line
Lecture 1
parallelism
not parallel
parallel
not parallel
not parallel
parallel
-
co-planar?
yes
yes
no
no
no
2014
Space
line &
plane
2 lines
Configurations of pair of lines
pair of objects
intersecting lines
parallel lines
skew lines
line intersecting plane
line parallel to a plane
line lying in plane
Math 141
intersection
one point
empty
none
one point
none
line
Lecture 1
parallelism
not parallel
parallel
not parallel
not parallel
parallel
-
co-planar?
yes
yes
no
no
no
?
2014
Space
line &
plane
2 lines
Configurations of pair of lines
pair of objects
intersecting lines
parallel lines
skew lines
line intersecting plane
line parallel to a plane
line lying in plane
Math 141
intersection
one point
empty
none
one point
none
line
Lecture 1
parallelism
not parallel
parallel
not parallel
not parallel
parallel
-
co-planar?
yes
yes
no
no
no
yes
2014
Space
line &
2 planes plane
2 lines
Configurations of pair of lines
pair of objects
intersecting lines
parallel lines
skew lines
line intersecting plane
line parallel to a plane
line lying in plane
intersecting planes
parallel planes
Math 141
intersection
one point
empty
none
one point
none
line
?
Lecture 1
parallelism
not parallel
parallel
not parallel
not parallel
parallel
-
co-planar?
yes
yes
no
no
no
yes
-
2014
Space
line &
2 planes plane
2 lines
Configurations of pair of lines
pair of objects
intersecting lines
parallel lines
skew lines
line intersecting plane
line parallel to a plane
line lying in plane
intersecting planes
parallel planes
Math 141
intersection
one point
empty
none
one point
none
line
line
Lecture 1
parallelism
not parallel
parallel
not parallel
not parallel
parallel
-
co-planar?
yes
yes
no
no
no
yes
-
2014
Space
line &
2 planes plane
2 lines
Configurations of pair of lines
pair of objects
intersecting lines
parallel lines
skew lines
line intersecting plane
line parallel to a plane
line lying in plane
intersecting planes
parallel planes
Math 141
intersection
one point
empty
none
one point
none
line
line
Lecture 1
parallelism
not parallel
parallel
not parallel
not parallel
parallel
?
co-planar?
yes
yes
no
no
no
yes
-
2014
Space
line &
2 planes plane
2 lines
Configurations of pair of lines
pair of objects
intersecting lines
parallel lines
skew lines
line intersecting plane
line parallel to a plane
line lying in plane
intersecting planes
parallel planes
Math 141
intersection
one point
empty
none
one point
none
line
line
Lecture 1
parallelism
not parallel
parallel
not parallel
not parallel
parallel
not parallel
co-planar?
yes
yes
no
no
no
yes
-
2014
Space
line &
2 planes plane
2 lines
Configurations of pair of lines
pair of objects
intersecting lines
parallel lines
skew lines
line intersecting plane
line parallel to a plane
line lying in plane
intersecting planes
parallel planes
Math 141
intersection
one point
empty
none
one point
none
line
line
?
Lecture 1
parallelism
not parallel
parallel
not parallel
not parallel
parallel
not parallel
co-planar?
yes
yes
no
no
no
yes
-
2014
Space
line &
2 planes plane
2 lines
Configurations of pair of lines
pair of objects
intersecting lines
parallel lines
skew lines
line intersecting plane
line parallel to a plane
line lying in plane
intersecting planes
parallel planes
Math 141
intersection
one point
empty
none
one point
none
line
line
none
Lecture 1
parallelism
not parallel
parallel
not parallel
not parallel
parallel
not parallel
co-planar?
yes
yes
no
no
no
yes
-
2014
Space
line &
2 planes plane
2 lines
Configurations of pair of lines
pair of objects
intersecting lines
parallel lines
skew lines
line intersecting plane
line parallel to a plane
line lying in plane
intersecting planes
parallel planes
Math 141
intersection
one point
empty
none
one point
none
line
line
none
Lecture 1
parallelism
not parallel
parallel
not parallel
not parallel
parallel
not parallel
?
co-planar?
yes
yes
no
no
no
yes
-
2014
Space
line &
2 planes plane
2 lines
Configurations of pair of lines
pair of objects
intersecting lines
parallel lines
skew lines
line intersecting plane
line parallel to a plane
line lying in plane
intersecting planes
parallel planes
Math 141
intersection
one point
empty
none
one point
none
line
line
none
Lecture 1
parallelism
not parallel
parallel
not parallel
not parallel
parallel
not parallel
parallel
co-planar?
yes
yes
no
no
no
yes
-
2014
Space
Distance
Euclidean Plane:
Through a point P outside a line L
there passes at most one line ` parallel to L
Math 141
Lecture 1
2014
Space
Distance
Euclidean Plane:
Through a point P outside a line L
there passes at most one line ` parallel to L
Distance - Primordial concept
Quantifies/Measures how close/far apart are any two points
d(A, B) = |AB|
Math 141
Lecture 1
2014
Space
Distance
Euclidean Plane:
Through a point P outside a line L
there passes at most one line ` parallel to L
Distance - Primordial concept
Quantifies/Measures how close/far apart are any two points
d(A, B) = |AB|
Measures of angles → Perpendicularity
Line and Line
Line and Plane
Plane and Plane
Math 141
Lecture 1
2014
Space
P2
L2
L1
L
P1
The planes P1 and P2 are perpendicular on each other;
The lines L2 and L are coplanar and perpendicular on each other;
The lines L1 and L2 are skew but perpendicular on each other;
The lines L1 and L are coplanar and not perpendicular;
The line L2 is perpendicular to the plane P1 ;
The line L1 is not perpendicular to the plane P2 .
Math 141
Lecture 1
2014
Space
Rectangular/Cartesian Coordinates
z
y
O
x
Math 141
A Cartesian coordinate system is given by
fixing:
Lecture 1
2014
Space
Rectangular/Cartesian Coordinates
z
y
O
x
Math 141
A Cartesian coordinate system is given by
fixing:
a point O (called the origin),
Lecture 1
2014
Space
Rectangular/Cartesian Coordinates
z
y
O
x
Math 141
A Cartesian coordinate system is given by
fixing:
a point O (called the origin),
3 pairwise perpendicular lines intersecting
at the origin,
Lecture 1
2014
Space
Rectangular/Cartesian Coordinates
z
y
O
x
Math 141
A Cartesian coordinate system is given by
fixing:
a point O (called the origin),
3 pairwise perpendicular lines intersecting
at the origin,
a direction in each of the coordinate axis.
Lecture 1
2014
Space
Rectangular/Cartesian Coordinates
z
y
O
x
A Cartesian coordinate system is given by
fixing:
a point O (called the origin),
3 pairwise perpendicular lines intersecting
at the origin,
a direction in each of the coordinate axis.
The three lines are labeled as x-axis, y -axis
and z-axis.
Math 141
Lecture 1
2014
Space
Rectangular/Cartesian Coordinates
P(xP , yP , zP )
z
P -point. We assign to it triple (xP , yP , zP ).
y
O
x
Math 141
Lecture 1
2014
Space
Rectangular/Cartesian Coordinates
P(xP , yP , zP )
z
P -point. We assign to it triple (xP , yP , zP ).
Assignment will be such that distinct points
are assigned distinct triples.
y
O
x
Math 141
Lecture 1
2014
Space
Rectangular/Cartesian Coordinates
P(xP , yP , zP )
z
P -point. We assign to it triple (xP , yP , zP ).
Assignment will be such that distinct points
are assigned distinct triples.
y
Q = base of perpendicular from P to x-axis.
O
x
Math 141
Lecture 1
2014
Space
Rectangular/Cartesian Coordinates
P(xP , yP , zP )
z
P -point. We assign to it triple (xP , yP , zP ).
Assignment will be such that distinct points
are assigned distinct triples.
y
Q = base of perpendicular from P to x-axis.
O
xP
x
Math 141
Define xP as signed distance b-n O and Q.
Lecture 1
2014
Space
Rectangular/Cartesian Coordinates
P(xP , yP , zP )
z
P -point. We assign to it triple (xP , yP , zP ).
Assignment will be such that distinct points
are assigned distinct triples.
y
Q = base of perpendicular from P to x-axis.
O
xP
x
Define xP as signed distance b-n O and Q.
Take distance with + sign if OQ points in
direction of x-axis, − sign else.
Math 141
Lecture 1
2014
Space
Rectangular/Cartesian Coordinates
P(xP , yP , zP )
z
P -point. We assign to it triple (xP , yP , zP ).
Assignment will be such that distinct points
are assigned distinct triples.
zP
yP
y
Q = base of perpendicular from P to x-axis.
O
xP
x
Define xP as signed distance b-n O and Q.
Take distance with + sign if OQ points in
direction of x-axis, − sign else.
Definitions of yP , zP are similar.
Math 141
Lecture 1
2014
Space
Rectangular/Cartesian Coordinates
P(xP , yP , zP )
z
P -point. We assign to it triple (xP , yP , zP ).
Assignment will be such that distinct points
are assigned distinct triples.
zP
yP
y
Q = base of perpendicular from P to x-axis.
O
xP
x
Define xP as signed distance b-n O and Q.
Take distance with + sign if OQ points in
direction of x-axis, − sign else.
Definitions of yP , zP are similar.
(xP , yP , zP ) = Cartesian coordinates of P.
Math 141
Lecture 1
2014
Space
Rectangular/Cartesian Coordinates
P(xP , yP , zP )
z
P -point. We assign to it triple (xP , yP , zP ).
Assignment will be such that distinct points
are assigned distinct triples.
zP
yP
y
Q = base of perpendicular from P to x-axis.
O
xP
x
Define xP as signed distance b-n O and Q.
Take distance with + sign if OQ points in
direction of x-axis, − sign else.
Definitions of yP , zP are similar.
(xP , yP , zP ) = Cartesian coordinates of P.
xP is called the x-coordinate of P, and so on
for other axes.
Math 141
Lecture 1
2014
Space
Rectangular/Cartesian Coordinates
P(xP , yP , zP )
z
P -point. We assign to it triple (xP , yP , zP ).
Assignment will be such that distinct points
are assigned distinct triples.
zP
yP
y
Q = base of perpendicular from P to x-axis.
O
xP
x
Define xP as signed distance b-n O and Q.
Take distance with + sign if OQ points in
direction of x-axis, − sign else.
Definitions of yP , zP are similar.
(xP , yP , zP ) = Cartesian coordinates of P.
xP is called the x-coordinate of P, and so on
for other axes.
(xP , yP , zP ) = singed lengths of edges of the
rectangular box indicated in the picture.
Math 141
Lecture 1
2014
Space
Euclidean Distance in Coordinates
Theorem (Can be taken as definition)
The distance b-n the
q points A(xA , yA , zA ) and B(xB , yB , zB ) is given by:
d(A, B) = |AB| =
(xB − xA )2 + (yB − yA )2 + (zB − zA )2
B
z
A
y
x
Math 141
Lecture 1
2014
Space
Euclidean Distance in Coordinates
Theorem (Can be taken as definition)
The distance b-n the
q points A(xA , yA , zA ) and B(xB , yB , zB ) is given by:
d(A, B) = |AB| =
(xB − xA )2 + (yB − yA )2 + (zB − zA )2
Why is this so? Geometric explanation:
B
z
C
A
y
D
x
Math 141
Lecture 1
2014
Space
Euclidean Distance in Coordinates
Theorem (Can be taken as definition)
The distance b-n the
q points A(xA , yA , zA ) and B(xB , yB , zB ) is given by:
d(A, B) = |AB| =
(xB − xA )2 + (yB − yA )2 + (zB − zA )2
Why is this so? Geometric explanation:
B
z
|AC|2 = |AD|2 + |DC|2
4ADC
C
A
y
D
x
Math 141
Lecture 1
2014
Space
Euclidean Distance in Coordinates
Theorem (Can be taken as definition)
The distance b-n the
q points A(xA , yA , zA ) and B(xB , yB , zB ) is given by:
d(A, B) = |AB| =
(xB − xA )2 + (yB − yA )2 + (zB − zA )2
Why is this so? Geometric explanation:
B
z
|AC|2 = |AD|2 + |DC|2
|AB|2 = |BC|2 + |AC|2
4ADC
4ACB
C
A
y
D
x
Math 141
Lecture 1
2014
Space
Euclidean Distance in Coordinates
Theorem (Can be taken as definition)
The distance b-n the
q points A(xA , yA , zA ) and B(xB , yB , zB ) is given by:
d(A, B) = |AB| =
(xB − xA )2 + (yB − yA )2 + (zB − zA )2
Why is this so? Geometric explanation:
B
z
C
A
y
|AC|2 = |AD|2 + |DC|2
|AB|2 = |BC|2 + |AC|2
= |BC|2 + |AD|2 + |DC|2
4ADC
4ACB
D
x
Math 141
Lecture 1
2014
Space
Euclidean Distance in Coordinates
Theorem (Can be taken as definition)
The distance b-n the
q points A(xA , yA , zA ) and B(xB , yB , zB ) is given by:
d(A, B) = |AB| =
(xB − xA )2 + (yB − yA )2 + (zB − zA )2
Why is this so? Geometric explanation:
B
z
C
A xB − xA
D
y
x
Math 141
|AC|2 = |AD|2 + |DC|2
|AB|2 = |BC|2 + |AC|2
= |BC|2 + |AD|2 + |DC|2
= (xB − xA )2 + (yB − yA )2
+(zB − zA )2 ,
Lecture 1
2014
4ADC
4ACB
Space
Euclidean Distance in Coordinates
Theorem (Can be taken as definition)
The distance b-n the
q points A(xA , yA , zA ) and B(xB , yB , zB ) is given by:
d(A, B) = |AB| =
(xB − xA )2 + (yB − yA )2 + (zB − zA )2
Why is this so? Geometric explanation:
B
z
A xB − xA
D
y
x
Math 141
C
yB − yA
|AC|2 = |AD|2 + |DC|2
|AB|2 = |BC|2 + |AC|2
= |BC|2 + |AD|2 + |DC|2
= (xB − xA )2 + (yB − yA )2
+(zB − zA )2 ,
Lecture 1
2014
4ADC
4ACB
Space
Euclidean Distance in Coordinates
Theorem (Can be taken as definition)
The distance b-n the
q points A(xA , yA , zA ) and B(xB , yB , zB ) is given by:
d(A, B) = |AB| =
(xB − xA )2 + (yB − yA )2 + (zB − zA )2
Why is this so? Geometric explanation:
B
z
zB − zA
A xB − xA
D
y
x
Math 141
C
yB − yA
|AC|2 = |AD|2 + |DC|2
|AB|2 = |BC|2 + |AC|2
= |BC|2 + |AD|2 + |DC|2
= (xB − xA )2 + (yB − yA )2
+(zB − zA )2 ,
Lecture 1
2014
4ADC
4ACB
Space
Euclidean Distance in Coordinates
Theorem (Can be taken as definition)
The distance b-n the
q points A(xA , yA , zA ) and B(xB , yB , zB ) is given by:
d(A, B) = |AB| =
(xB − xA )2 + (yB − yA )2 + (zB − zA )2
Why is this so? Geometric explanation:
B
z
zB − zA
A xB − xA
D
y
x
C
yB − yA
|AC|2 = |AD|2 + |DC|2
|AB|2 = |BC|2 + |AC|2
= |BC|2 + |AD|2 + |DC|2
= (xB − xA )2 + (yB − yA )2
+(zB − zA )2 ,
4ADC
4ACB
Example:
d(P(3, 1, 2), Q(1, 2, 3)) = ?
Math 141
.
Lecture 1
2014
Space
Euclidean Distance in Coordinates
Theorem (Can be taken as definition)
The distance b-n the
q points A(xA , yA , zA ) and B(xB , yB , zB ) is given by:
d(A, B) = |AB| =
(xB − xA )2 + (yB − yA )2 + (zB − zA )2
Why is this so? Geometric explanation:
B
z
zB − zA
A xB − xA
D
y
C
yB − yA
x
|AC|2 = |AD|2 + |DC|2
|AB|2 = |BC|2 + |AC|2
= |BC|2 + |AD|2 + |DC|2
= (xB − xA )2 + (yB − yA )2
+(zB − zA )2 ,
4ADC
4ACB
Example:
d(P(3, 1, 2), Q(1, 2, 3)) =
Math 141
q
√
(1 − 3)2 + (2 − 1)2 + (3 − 2)2 = 6 .
Lecture 1
2014
Space
Sets in Space
X subset of a set Y :
X = {A in Y |A has property P} ⊂ Y
Math 141
Lecture 1
2014
Space
Sets in Space
X subset of a set Y :
X = {A in Y |A has property P} ⊂ Y
Examples (Fixed point Q, fixed r > 0):
X = {A in Space |d(A, Q) = r } = Sr (Q) ,
Math 141
Lecture 1
2014
Space
Sets in Space
X subset of a set Y :
X = {A in Y |A has property P} ⊂ Y
Examples (Fixed point Q, fixed r > 0):
X = {A in Space |d(A, Q) = r } = Sr (Q) ,
Sphere of radius r centered at Q.
Math 141
Lecture 1
2014
Space
Sets in Space
X subset of a set Y :
X = {A in Y |A has property P} ⊂ Y
Examples (Fixed point Q, fixed r > 0):
X = {A in Space |d(A, Q) = r } = Sr (Q) ,
Sphere of radius r centered at Q.
Br (Q) = {A in Space |d(A, Q) < r } ,
Math 141
Lecture 1
2014
Space
Sets in Space
X subset of a set Y :
X = {A in Y |A has property P} ⊂ Y
Examples (Fixed point Q, fixed r > 0):
X = {A in Space |d(A, Q) = r } = Sr (Q) ,
Sphere of radius r centered at Q.
Br (Q) = {A in Space |d(A, Q) < r } ,
Open ball of radius r centered at Q.
Math 141
Lecture 1
2014
Space
Sets in Space
X subset of a set Y :
X = {A in Y |A has property P} ⊂ Y
Examples (Fixed point Q, fixed r > 0):
X = {A in Space |d(A, Q) = r } = Sr (Q) ,
Sphere of radius r centered at Q.
Br (Q) = {A in Space |d(A, Q) < r } ,
Open ball of radius r centered at Q.
B r (Q) = {A in Space |d(A, Q) 6 r } ,
Math 141
Lecture 1
2014
Space
Sets in Space
X subset of a set Y :
X = {A in Y |A has property P} ⊂ Y
Examples (Fixed point Q, fixed r > 0):
X = {A in Space |d(A, Q) = r } = Sr (Q) ,
Sphere of radius r centered at Q.
Br (Q) = {A in Space |d(A, Q) < r } ,
Open ball of radius r centered at Q.
B r (Q) = {A in Space |d(A, Q) 6 r } ,
Closed ball of radius r centered at Q.
Math 141
Lecture 1
2014
Space
Equation(s) of Subsets
X = {(x, y , z)|x, y , z satisfy certain relation(s) } .
Math 141
Lecture 1
2014
Space
Equation(s) of Subsets
X = {(x, y , z)|x, y , z satisfy certain relation(s) } .
Examples:
{(x, y , z)|x 2 + y 2 + z 2 = 1}:
Math 141
Lecture 1
2014
Space
Equation(s) of Subsets
X = {(x, y , z)|x, y , z satisfy certain relation(s) } .
Examples:
{(x, y , z)|x 2 + y 2 + z 2 = 1}:
sphere of radius r = 1 centered at the origin (0, 0, 0)
Also refered to as: sphere x 2 + y 2 + z 2 = 1
Math 141
Lecture 1
2014
Space
Equation(s) of Subsets
X = {(x, y , z)|x, y , z satisfy certain relation(s) } .
Examples:
{(x, y , z)|x 2 + y 2 + z 2 = 1}:
sphere of radius r = 1 centered at the origin (0, 0, 0)
Also refered to as: sphere x 2 + y 2 + z 2 = 1
{(x, y , z)|x = 0}: coordinate Left-Up plane
Math 141
Lecture 1
2014
Space
Equation(s) of Subsets
X = {(x, y , z)|x, y , z satisfy certain relation(s) } .
Examples:
{(x, y , z)|x 2 + y 2 + z 2 = 1}:
sphere of radius r = 1 centered at the origin (0, 0, 0)
Also refered to as: sphere x 2 + y 2 + z 2 = 1
{(x, y , z)|x = 0}: coordinate Left-Up plane
{(x, y , z)|x = 0 and y = 0}:
intersection of coordinate planes → coordinate axis
Math 141
Lecture 1
2014
Space
Equation(s) of Subsets
X = {(x, y , z)|x, y , z satisfy certain relation(s) } .
Examples:
{(x, y , z)|x 2 + y 2 + z 2 = 1}:
sphere of radius r = 1 centered at the origin (0, 0, 0)
Also refered to as: sphere x 2 + y 2 + z 2 = 1
{(x, y , z)|x = 0}: coordinate Left-Up plane
{(x, y , z)|x = 0 and y = 0}:
intersection of coordinate planes → coordinate axis
Can be given by only one equation:
x 2 + y 2 = 0 → x = 0,y = 0, and z arbitrary →
vertical axis above (0, 0) in (x, y )−plane
Math 141
Lecture 1
2014
Space
Equation(s) of Subsets
X = {(x, y , z)|x, y , z satisfy certain relation(s) } .
Examples:
{(x, y , z)|x 2 + y 2 + z 2 = 1}:
sphere of radius r = 1 centered at the origin (0, 0, 0)
Also refered to as: sphere x 2 + y 2 + z 2 = 1
{(x, y , z)|x = 0}: coordinate Left-Up plane
{(x, y , z)|x = 0 and y = 0}:
intersection of coordinate planes → coordinate axis
Can be given by only one equation:
x 2 + y 2 = 0 → x = 0,y = 0, and z arbitrary →
vertical axis above (0, 0) in (x, y )−plane
Important: Equations in Plane vs. Space.
Math 141
Lecture 1
2014
Space
Recognizing Spheres from Equations
Q(x0 , y0 , z0 ), r > 0, A(x, y , z). Remark: d(A, Q) = r ←→ d 2 (A, Q) = r 2
Sr (Q) :
Math 141
(x − x0 )2 + (y − y0 )2 + (z − z0 )2 = r 2
Lecture 1
2014
Space
Recognizing Spheres from Equations
Q(x0 , y0 , z0 ), r > 0, A(x, y , z). Remark: d(A, Q) = r ←→ d 2 (A, Q) = r 2
Sr (Q) :
(x − x0 )2 + (y − y0 )2 + (z − z0 )2 = r 2
Example:
(x − 2)2 + (y − 0)2 + (z + 1)2 = 32
x 2 + y 2 + z 2 − 4x + 2z − 4 = 0
Math 141
Lecture 1
2014
Space
Recognizing Spheres from Equations
Q(x0 , y0 , z0 ), r > 0, A(x, y , z). Remark: d(A, Q) = r ←→ d 2 (A, Q) = r 2
Sr (Q) :
(x − x0 )2 + (y − y0 )2 + (z − z0 )2 = r 2
Example:
(x − 2)2 + (y − 0)2 + (z + 1)2 = 32
x 2 + y 2 + z 2 − 4x + 2z − 4 = 0
no mixed terms xy , xz, or yz;
quadratic terms x 2 , y 2 , and z 2 with the same coefficient.
Math 141
Lecture 1
2014
Space
Recognizing Spheres from Equations
Q(x0 , y0 , z0 ), r > 0, A(x, y , z). Remark: d(A, Q) = r ←→ d 2 (A, Q) = r 2
Sr (Q) :
(x − x0 )2 + (y − y0 )2 + (z − z0 )2 = r 2
Example:
(x − 2)2 + (y − 0)2 + (z + 1)2 = 32
x 2 + y 2 + z 2 − 4x + 2z − 4 = 0
no mixed terms xy , xz, or yz;
quadratic terms x 2 , y 2 , and z 2 with the same coefficient.
Examples:
x 2 + y 2 + z 2 − 4x + 2y = 0
Math 141
Lecture 1
2014
Space
Recognizing Spheres from Equations
Q(x0 , y0 , z0 ), r > 0, A(x, y , z). Remark: d(A, Q) = r ←→ d 2 (A, Q) = r 2
Sr (Q) :
(x − x0 )2 + (y − y0 )2 + (z − z0 )2 = r 2
Example:
(x − 2)2 + (y − 0)2 + (z + 1)2 = 32
x 2 + y 2 + z 2 − 4x + 2z − 4 = 0
no mixed terms xy , xz, or yz;
quadratic terms x 2 , y 2 , and z 2 with the same coefficient.
Examples:
x 2 + y 2 + z 2 − 4x + 2y = 0
Complete the square:
Sphere of radius
Math 141
√
(x − 2)2 + (y + 1)2 + z 2 = 5
5 centered at (2, −1, 0).
Lecture 1
2014
Space
Recognizing Spheres from Equations
Q(x0 , y0 , z0 ), r > 0, A(x, y , z). Remark: d(A, Q) = r ←→ d 2 (A, Q) = r 2
Sr (Q) :
(x − x0 )2 + (y − y0 )2 + (z − z0 )2 = r 2
Example:
(x − 2)2 + (y − 0)2 + (z + 1)2 = 32
x 2 + y 2 + z 2 − 4x + 2z − 4 = 0
no mixed terms xy , xz, or yz;
quadratic terms x 2 , y 2 , and z 2 with the same coefficient.
Examples:
x 2 + y 2 + z 2 − 4x + 2y = 0
Complete the square:
(x − 2)2 + (y + 1)2 + z 2 = 5
√
Sphere of radius 5 centered at (2, −1, 0).
How about x 2 + y 2 + z 2 − 4x + 2y = −6?
Math 141
Lecture 1
2014
Space
Recognizing Spheres from Equations
Q(x0 , y0 , z0 ), r > 0, A(x, y , z). Remark: d(A, Q) = r ←→ d 2 (A, Q) = r 2
Sr (Q) :
(x − x0 )2 + (y − y0 )2 + (z − z0 )2 = r 2
Example:
(x − 2)2 + (y − 0)2 + (z + 1)2 = 32
x 2 + y 2 + z 2 − 4x + 2z − 4 = 0
no mixed terms xy , xz, or yz;
quadratic terms x 2 , y 2 , and z 2 with the same coefficient.
Examples:
x 2 + y 2 + z 2 − 4x + 2y = 0
Complete the square:
(x − 2)2 + (y + 1)2 + z 2 = 5
√
Sphere of radius 5 centered at (2, −1, 0).
How about x 2 + y 2 + z 2 − 4x + 2y = −6? Passes both tests, but ...
Math 141
Lecture 1
2014