a rule that combines two vectors to produce a scalar
TRANSCRIPT
A rule that combines two vectors to produce a scalar
2
7
1
2
4
2
5
3
6 + -5 + -14 + 8
= -5
n
iii
nn
ba
b
b
b
a
a
a
1
2
1
2
1
y
xv
x
y
22 yxy
x
y
xvv
y
xv
x
y
22 yxy
x
y
xvv
= NORM of v
u
v
v
)( vu vu
u
v
vu
u
v
vu
the norm of the vector is vu )( vu )( vu
vuvvuu 2
The SQUARE of
u
v
vu
the norm of the vector is vu vuvvuu 2
a
b
c
The SQUARE of
law of cosines:
c2 = a2 + b2 - 2abcos
coscos vuabvu
cosvuvu
cosvu
vu
In 2 or 3 dimensional space, the vectors u and v are perpendicular if the angle between them ( ie ) is 90 degrees.
090cos
vu
vuiff
0vu
definition: two vectors are said to be ORTHOGONALif and only if their dot product is zero.
4149
15412
5326
1
1
2
3
4123
1524
5312
3 4 matrix vector in R4
vector in R3
M v
4149
15412
5326
1
1
2
3
4123
1524
5312
M v
The first entry
is the first row of M dot v
2
4149
15412
5326
1
1
2
3
4123
1524
5312
4149
15412
5326
1
1
2
3
4123
1524
5312
M v
The second entry
is the second row of M dot v
4149
15412
5326
1
1
2
3
4123
1524
5312
M v
10
4149
15412
5326
1
1
2
3
4123
1524
5312
M v
The third entry
is the third row of M dot v
4149
15412
5326
1
1
2
3
4123
1524
5312
M v
0
0
10
2
4149
15412
5326
1
1
2
3
4123
1524
5312
M v
Consider some alternate ways of describing the following system:
1211109
8765
4321
zyx
zyx
zyx
1211109
8765
4321
zyx
zyx
zyx
9
5
1
10
6
2
11
7
3
12
8
4
x y z+ + =
1211109
8765
4321
zyx
zyx
zyx
9
5
1
10
6
2
11
7
3
1211109
8765
4321
zyx
zyx
zyx
9
5
1
10
6
2
11
7
3
12
8
4
z
y
x
=
1211109
8765
4321
zyx
zyx
zyx
9
5
1
10
6
2
11
7
3
12
8
4
z
y
x
=
9
5
1
10
6
2
11
7
3
12
8
4
x y z+ + =