vectors Euclidean vector spaces
A rector is represented by an arrow 2 space 3 space
- magnitude and direction
3
all ordered
IR
vectors in 2-space 3-space and n-space
,
lists with 2
elements in the
& (2, 3)
d (a , b , 3)
is a quantity form of a list
set of all
(a , b)
I &
"list" 1112,3) lists that "octants"
2
are n long
E IR2
3) (a , b , c
... n)
l -"M
A3 IRh ↑
He
N
is a subset of & representations
a vector is
He is not a subset because any list of things
it is made up of different That represents a
AX , DY
things concept
RKIR2 u <4 ,3) represents
4
=
a
an ordered pair is different ~ change
man an ordered triple
ic
min
Space
initial
↳3
matrix = addition addition
rector of vectors is commutative
+i = u i +h =
multiply by
A (3, 4) B = ( 3 , 10)
·
-
=
Scalars
<6 , 1475
3
no vector
a
resultant mutiplication
terminal-initial
i = (a , b)
cos(r)
unit vector
VII Nb2
-=
in
=
+
In R = IV11-
dot product is a measure
of agreement
11k F 1 : MV I = 90
d
magnitude direction normal orthogonal
⑦ dot products agree
↓
kid "unrelated" O dot products :
disagree
,orthogonality
Normal Orthogonal 3X + 4y -
z = 12
"related ·
10 , 0 , -12)
(4 , 0 , 0
-
10 , 3 , 01
integers
a(x Xo) +b(yy0) +
c(z 20)
-
0
(40
-
=
, 107 < 3, -2 47
,
# an infinite #
⑤
&
, -416
10
M
↑
II
20 ,4 , 27
of XS
y 25
Sa -
2b +4c = 0 points
(2 , 5 07 =+G 8 orthogonal
4
,
-
, O to
I it
7-4 , 0 5)
, 10- Litsa do+ product!
10
,
8 47
, proje = Malco
<4 10 0
, ,
1-8 ,
0 ,
107
20 ,
2
, K
&
,projections
↓ projab at+ b= c - projet +e
=
b
The amount one rector "the rest of h
"
Iprojdist
written
3
goes in the direction of as sum of
another vector - between vector parallel
.
projab and a and
to
orthogonal to
be T
magnitude IlbII cost "onto" Vector
proje
-
=
the
projection of b snows up
-
direction of ↓ Onto a a 10t
>
-
projet = 11511coso . -
EX .
proj : 1 , 01 just magnitude
a = 3i + 2j 4k -
+ <3 , 2
,
-
4)
↓ " proje =
b = i bk-
<1 , 0 ,6)
+
, -
from
0
b a -
= 3+ 0 + 24 = 27
N
(8/10)
-
a (2, 1 , 1)
proj(-proj
-
=
↑fi
5 =
b = (1 , 1 5)
-
⑮ L-
proje d
ja 2+ 1 + 5
+ (45)-
-
18 , .
zu
, cross products ONLY APPLIES TO R3
u = < 1 , 2 , 3) = <4 , 5 , 67
=
If <U ,, 42 , Ug) and <V Ve Vs) , , ⑫
( I
i jk
xi123
then uxu is the rector defined
456
=
1581 il) -
+
by <
UzVz-UzVe , UgV,
-
U , Vz U ,
, Vz-UzV1] k(45
Differences -
3i +
6j -
3k perpendicular
(ix]
=
. i = 0 to u and
① V x axb v
n&
= Scalar = Vector
(x) v 0 .
=
↑
②. = vi .
uxv = -
([X()
③. =
0 V x 0I =
u
3 oi oj +o
+
volume of a
magnitude area of
~
of a parallelogram- parallelepiped
parallelogram
XVll = 11411 II VII sing
# V= b h .
=
↓
COSO multiply proju t
proje.
U V
.
= 11411 /IVII luxwil
cross : sn
o cos" (ii) simplified :
↓
=
EX
/(bx))
.
V= x
1) uXvll = 1411 II VIISinG
· paradetopiped
↓
-
v n (rXw)
.
(x)
=
v n
/
=
tili-toilj /
.
uxv :
= + =
triple scalar
product
Ti
-
-
j + 3k = M49
1 9 + +
=
19
PQRS
(2,6) (11,1) (4 , 6,2)
op (1 2) (4 ,4) (13) ( ,3)
,
1
P Q R
pa pe LS76
PE &S
↑ & -
e
-Q
3773, 2)
↓ (2,0, 3) ,
(1, 5 , 2)
1300 ; -j)8
-
+
lluxul) = 11411 NVII Sing
.
2
21
1 = Gi -
Oj + 2)k
A rector is represented by an arrow 2 space 3 space
- magnitude and direction
3
all ordered
IR
vectors in 2-space 3-space and n-space
,
lists with 2
elements in the
& (2, 3)
d (a , b , 3)
is a quantity form of a list
set of all
(a , b)
I &
"list" 1112,3) lists that "octants"
2
are n long
E IR2
3) (a , b , c
... n)
l -"M
A3 IRh ↑
He
N
is a subset of & representations
a vector is
He is not a subset because any list of things
it is made up of different That represents a
AX , DY
things concept
RKIR2 u <4 ,3) represents
4
=
a
an ordered pair is different ~ change
man an ordered triple
ic
min
Space
initial
↳3
matrix = addition addition
rector of vectors is commutative
+i = u i +h =
multiply by
A (3, 4) B = ( 3 , 10)
·
-
=
Scalars
<6 , 1475
3
no vector
a
resultant mutiplication
terminal-initial
i = (a , b)
cos(r)
unit vector
VII Nb2
-=
in
=
+
In R = IV11-
dot product is a measure
of agreement
11k F 1 : MV I = 90
d
magnitude direction normal orthogonal
⑦ dot products agree
↓
kid "unrelated" O dot products :
disagree
,orthogonality
Normal Orthogonal 3X + 4y -
z = 12
"related ·
10 , 0 , -12)
(4 , 0 , 0
-
10 , 3 , 01
integers
a(x Xo) +b(yy0) +
c(z 20)
-
0
(40
-
=
, 107 < 3, -2 47
,
# an infinite #
⑤
&
, -416
10
M
↑
II
20 ,4 , 27
of XS
y 25
Sa -
2b +4c = 0 points
(2 , 5 07 =+G 8 orthogonal
4
,
-
, O to
I it
7-4 , 0 5)
, 10- Litsa do+ product!
10
,
8 47
, proje = Malco
<4 10 0
, ,
1-8 ,
0 ,
107
20 ,
2
, K
&
,projections
↓ projab at+ b= c - projet +e
=
b
The amount one rector "the rest of h
"
Iprojdist
written
3
goes in the direction of as sum of
another vector - between vector parallel
.
projab and a and
to
orthogonal to
be T
magnitude IlbII cost "onto" Vector
proje
-
=
the
projection of b snows up
-
direction of ↓ Onto a a 10t
>
-
projet = 11511coso . -
EX .
proj : 1 , 01 just magnitude
a = 3i + 2j 4k -
+ <3 , 2
,
-
4)
↓ " proje =
b = i bk-
<1 , 0 ,6)
+
, -
from
0
b a -
= 3+ 0 + 24 = 27
N
(8/10)
-
a (2, 1 , 1)
proj(-proj
-
=
↑fi
5 =
b = (1 , 1 5)
-
⑮ L-
proje d
ja 2+ 1 + 5
+ (45)-
-
18 , .
zu
, cross products ONLY APPLIES TO R3
u = < 1 , 2 , 3) = <4 , 5 , 67
=
If <U ,, 42 , Ug) and <V Ve Vs) , , ⑫
( I
i jk
xi123
then uxu is the rector defined
456
=
1581 il) -
+
by <
UzVz-UzVe , UgV,
-
U , Vz U ,
, Vz-UzV1] k(45
Differences -
3i +
6j -
3k perpendicular
(ix]
=
. i = 0 to u and
① V x axb v
n&
= Scalar = Vector
(x) v 0 .
=
↑
②. = vi .
uxv = -
([X()
③. =
0 V x 0I =
u
3 oi oj +o
+
volume of a
magnitude area of
~
of a parallelogram- parallelepiped
parallelogram
XVll = 11411 II VII sing
# V= b h .
=
↓
COSO multiply proju t
proje.
U V
.
= 11411 /IVII luxwil
cross : sn
o cos" (ii) simplified :
↓
=
EX
/(bx))
.
V= x
1) uXvll = 1411 II VIISinG
· paradetopiped
↓
-
v n (rXw)
.
(x)
=
v n
/
=
tili-toilj /
.
uxv :
= + =
triple scalar
product
Ti
-
-
j + 3k = M49
1 9 + +
=
19
PQRS
(2,6) (11,1) (4 , 6,2)
op (1 2) (4 ,4) (13) ( ,3)
,
1
P Q R
pa pe LS76
PE &S
↑ & -
e
-Q
3773, 2)
↓ (2,0, 3) ,
(1, 5 , 2)
1300 ; -j)8
-
+
lluxul) = 11411 NVII Sing
.
2
21
1 = Gi -
Oj + 2)k
