minate se
a
Y Ist quadrant
In 2D
IR2 =
((X y)(x
, ,
y <R)
y f(x)
=
.
* *
projection
Distance
of
from
P on
p
to
to
by y z
xy plane
place
=
expo a
(x
10
z
,
,
0
y z)
,
,
t)
* Equations of geometric objects
1-dim
in R
S Y
XZ
z
=
=
Y
X
objects lines
-
:
curves
·
coordinate axis
X free
* Distance from P to x axis =
I
vas
X axis (x = x)
y
=
z =
-
: x
Y (
=
+ z2
y - axim : x = z = w
(y y)
=
z-axis : X =
Y = w (z = z( Z
=y
2
.
g * Distance from to
..
Origon p(0 0)
.
,
0
- ,
1041 =y+ z
* Distance between 4(X ) & Q (x2 Y2 ,
..
Y , , z 1 , = 2)
iPal : 42 x ( .
2
+ (y2 y-
. ) + (zz -
z ,
14
x+ 1 z
y 0
= =
,
Not a line
x+
y + z =
14
is a
plane
X=
y = z 7 A live
, I
#
2D objects surface places
-
:
,
·
coordinate
places
Free var
↓ X
xy place : z = 0 (x = X ,
j =
y)
X
yz = r
xZ y = 0
*
Equation of other places
2
. X = 2
x+ +z
y
=
1
2x +
3y + 5z =
6
·
surface in R3
e
.
g
.
sphere
centered O (0 0)
S
.
0 ,
Radius
..
=
r
10P1 = w
x= +
y + z = ra
centered 2[X0 %0 zo)
S
, ,
Radius = r
ICP1 = v
(x -
Xo("+ (y yo)+ (z zo)" ri- -
=
X
eg .
x+ 2
y+ 5z2
=
5 ( ellipsoid lit
2x2 +
2y2 + 222 = 8x -
4z + 2
2x2 22 +
8x +
2y2 + 42 2
=
2(X -
2) + 2y + 2(z + 1)2 = 12
(x-2)" y +
+ z + 1 = 6 < sphere centered at <(2 ,
0
. -1)
u = Vo
, I
#
# ea
#yo :8 pper wal space
ou
2
.
g
.
x2+ z-
v
, I
-
2
12 x +
y+ z* 4
Chap
~
12 2
. Vector
In 24
-
I
· A vector is a
object that specifies
& direction .
a
magnitude a
(length
rector IB : arrow from A to B
· same
magni & length
2
-
.
g. =
In 2D A (2 1) B (5 3)
, , ,
same rector
a =
B =
<5 -
2
,
3 -
1) =
33 , 2)
of & B are
different geometric
In 3D , P(1 ,
2 . 1) & & (2 · 3 , 1 representations of the same rector .
= =
Fr = (2 1
b (1))
-
2 1
2)
=
x
-
,
1
-
,
, ,
In 21 pts A(x1 y,) & B(X2, Y2)
given
·
,
,
The zero
-
rector : In 2D 5 =
10 , 03 therewith representation
a
>
-
In 3D5 a <Xz X,,
10 y )
=
= 0 6) ya
-
-
, , ,
/X
↑
Xcom - y com
#s the pation -
Dont confuse a
1)
pitas a
& (
a
Y Ist quadrant
In 2D
IR2 =
((X y)(x
, ,
y <R)
y f(x)
=
.
* *
projection
Distance
of
from
P on
p
to
to
by y z
xy plane
place
=
expo a
(x
10
z
,
,
0
y z)
,
,
t)
* Equations of geometric objects
1-dim
in R
S Y
XZ
z
=
=
Y
X
objects lines
-
:
curves
·
coordinate axis
X free
* Distance from P to x axis =
I
vas
X axis (x = x)
y
=
z =
-
: x
Y (
=
+ z2
y - axim : x = z = w
(y y)
=
z-axis : X =
Y = w (z = z( Z
=y
2
.
g * Distance from to
..
Origon p(0 0)
.
,
0
- ,
1041 =y+ z
* Distance between 4(X ) & Q (x2 Y2 ,
..
Y , , z 1 , = 2)
iPal : 42 x ( .
2
+ (y2 y-
. ) + (zz -
z ,
14
x+ 1 z
y 0
= =
,
Not a line
x+
y + z =
14
is a
plane
X=
y = z 7 A live
, I
#
2D objects surface places
-
:
,
·
coordinate
places
Free var
↓ X
xy place : z = 0 (x = X ,
j =
y)
X
yz = r
xZ y = 0
*
Equation of other places
2
. X = 2
x+ +z
y
=
1
2x +
3y + 5z =
6
·
surface in R3
e
.
g
.
sphere
centered O (0 0)
S
.
0 ,
Radius
..
=
r
10P1 = w
x= +
y + z = ra
centered 2[X0 %0 zo)
S
, ,
Radius = r
ICP1 = v
(x -
Xo("+ (y yo)+ (z zo)" ri- -
=
X
eg .
x+ 2
y+ 5z2
=
5 ( ellipsoid lit
2x2 +
2y2 + 222 = 8x -
4z + 2
2x2 22 +
8x +
2y2 + 42 2
=
2(X -
2) + 2y + 2(z + 1)2 = 12
(x-2)" y +
+ z + 1 = 6 < sphere centered at <(2 ,
0
. -1)
u = Vo
, I
#
# ea
#yo :8 pper wal space
ou
2
.
g
.
x2+ z-
v
, I
-
2
12 x +
y+ z* 4
Chap
~
12 2
. Vector
In 24
-
I
· A vector is a
object that specifies
& direction .
a
magnitude a
(length
rector IB : arrow from A to B
· same
magni & length
2
-
.
g. =
In 2D A (2 1) B (5 3)
, , ,
same rector
a =
B =
<5 -
2
,
3 -
1) =
33 , 2)
of & B are
different geometric
In 3D , P(1 ,
2 . 1) & & (2 · 3 , 1 representations of the same rector .
= =
Fr = (2 1
b (1))
-
2 1
2)
=
x
-
,
1
-
,
, ,
In 21 pts A(x1 y,) & B(X2, Y2)
given
·
,
,
The zero
-
rector : In 2D 5 =
10 , 03 therewith representation
a
>
-
In 3D5 a <Xz X,,
10 y )
=
= 0 6) ya
-
-
, , ,
/X
↑
Xcom - y com
#s the pation -
Dont confuse a
1)
pitas a
& (
