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+ + =