```                  Self Teaching Publications

VECTOR MATHEMATICS

a two and three dimensional algebra

by james n. hall

copyright © 1987 by James Norman Hall

United States of America

All Mathematical and Vector conventions as defined

in current math texts  are assumed.  The following small

presentation of VECTOR MATHEMATICS which is subtitled the One

Two and Three Dimensional Algebra does not challenge what has

been presently developed.  Several new definitions in VECTOR

MATHEMATICS should add another view point to Vector Analysis.

The main purpose of this treatise is to increase our understanding

of Algebra.  By introducing an expanded view point of the imaginary

operator "i" common to Algebraic Theory, an enhanced and extended

understanding of Algebra's connection with the real world is

established.

Let us begin our discussion of VECTOR MATHEMATICS by

discussing the general topic of the square root of any real

number.

Let "+/- r" be the square root of any real number "x"

or ( r )^2 = x or ( -r )^2 = x:  so long as a number

"r" can be found among the real number system.  If

x < 0, then no real number can be found in the real

number system. By convention the imaginary operator

"i" has been assumed to alleviate this dilemma. Therefore

( x )^(1/2) = ri if x < 0, where "i" is the imaginary

operator "i" which is defined as ( -1 )^(1/2) or

( i )^(1/2) = -1 or ( ri )^2 = x when ever x < 0.

Note: ( +i )^2 = -1 and ( -i )^2 = -1 also it is true

+i =/= -i (( =/= will be used to represent: NOT EQUAL TO ))

even though ( +i )^2 = ( -i )^2 = -1. Again to express

the imaginary operators properties in other forms,

( -1 )^.5 or ( -1 )^(1/2) = +i or -i = +/- i.

( x )^.5 = ri or ( ri )^2 = x when x < 0.

Again ( +i =/= -i ) means that +i + -i = 0.

What operators must be defined then when one takes the

square root of "i" iteritively? Or the square root of

the square root of the square root of the square root of

the square root of . . .  the square root of "i"?  That is

the "n"th level root of "i".

or

( . . . ( -1 )^.5 )^.5 )^.5 )^.5 )^.5 )^.5 . . . )^.5

or

( . . . ( -1 )^(1/2) )^(1/2) )^(1/2) )^(1/2) ) . . . )^(1/2)

or

( -1 )^(1/2n) as n --> t --> OO

or

( -1 )^(2n) as n --> t --> 0

where "t" is chosen as larger

than any other number T as many

times as desired. The constant "2"

is necessary to maintain an even

root of the  real number "-1".

Please note the important binary series characteristics

of the above treatment of successive root levels of the real

number "-1".  If "n" is an integer then the expression "(2n)"

creates the binary sequence of:

"0,2,4,8,16,32,64,128,256,512,1024,2048,4096, . . . 2n"

or the binary fraction of:

"1/2,1/4,1/8,1/16,1/32,1/64,1/128,1/256,1/512,1/1024,1/2048 . . . 1/2n"

The above two series suggest an application of the roots levels on the

imaginary operator to computer logic and machine level code.

(More will be discussed later concerning the use of binary relationships

with the imaginary operator "i" or any of its roots ( i )^(1/2n) for

any integer "n".)  For now let us continue our investigation of the roots

of "i".

Since ( -1 )^(1/2) or "i" is not in the real number system

then ( ( -1 )^(1/2) )^(1/2) or ( i )^(1/2) is not in the real

number system.  This applies to any root level also as the square

root  of "i" is the same as ( -1 )^(1/4).  The square root of

the square root of the square root of "i"  or in symbolic form

( i )^(1/8)  = ( -1 )^(1/16).  Let us define the root level of

"i" with the following convention:

let ( -1 )^(1/2) = i_2 , ( -1 )^(1/4) = i_4 ,

( -1 )^(1/8) = i_8

or

(  . . . ( -1 )^(1/2) )^(1/2) )^(1/2) . . . )^(1/2) = i_2n

where n in the number of root levels that "-1" is taken

to by the expression.

or in general:

( i_2n )^(2n) = -1

Multiplying the vector "A" or A_{x,y} or | a |_{x,y} by "-1"

rotates the vector pi radians or 180 degrees.  pi = 180 degrees.

or

( -1 ) * A_{x,y} = A_{x + pi,y + pi}

or

( -1 ) * A_{x,y} = A_{x + 180 degrees,y + 180 degrees}

In the above examples "x" represents not the x axis but

the angle A is from the x axis and likewise the "y" is

the angle to A from the y axis.  A_{x,y} = | a |_x,y

where "a" is the magnitude of the vector and "x" and "y"

are the angles from the x axis and the y axis respectively

Now ( ( i )^(1/2) )^2 = -1 therefore substituting in the

above expressions:

( ( i )^(1/2) )^2 * A_{x,y} = A_{x + pi,y + pi}

or

( ( i )^(1/2) )^2 * A_{x,y} = A_{x + 180 degrees,y + 180 degrees}

or

( i_2)^2 * A = A_{x + pi,y + pi}

Again if ( i_4 )^4 = -1 or ( i_n )^n = -1 then

( i_n )^n * A_{x,y} = A_{x + pi,y + pi}

Graphically represented see opposing  page figure 1.

If "-1" in all its various forms rotates  a vector 180

degrees or pi radians, then (+i_2) * (+i_2) or (-i_2) * (i_2)

also rotates the vector 180 degrees or pi radians as per

figure 2 on the opposing page in graphical form.

Definition:  "-1" rotates the vector A pi or 180 degrees.

then "+i" rotates the vector A "pi/2" or 90 degrees.  Also

"+i_4" or +( i )^(1/2) rotates the Vector A "pi/4" or 45 degrees.

Thus for every binary root level of the operator "i" the vector

A is rotated per the following table:

Vector "A" with magnitude "a" is rotated 360 degrees or 2pi

( -1 ) * A rotates A 180 degrees or pi radians counter

clockwise

( +i ) * A rotates A 90 degrees or  pi/2 radians counter

clockwise

( +i_4 ) * A rotates A 45 degrees or pi/4 radians counter

clockwise

( +i_8 ) * A rotates A 22.5 degrees or pi/8 radians counter

clockwise

( +i_32 ) * A rotates A 5.625 degrees or pi/32 radians

counter clockwise

( +i_64 ) * A rotates A 2.8125 degrees or pi/64 radians

counter clockwise

( +i_128 ) * A rotates A 1.40625  degrees or pi/128 radians

counter clockwise

( +i_(2^n) ) * A rotates A 180/(2^n) degrees or pi/(2^n)

radians counter clockwise for all integer values of "n".

And general for negative values of "i" the vector A

is rotated clockwise per the following rule:

( -i_(2^n) ) * A rotates A 180/(2^n) degrees or pi/(2^n)

radians  clockwise for all integer values of "n".

Example 1:

What root levels of "i" rotates a vector 30 degrees?

or find "q" such that A rotates 30 degrees

( q  ) * A_{w} = A_{w + 30 degrees}

( i_8 + i_32 + i_128 + i_512 + s ) * A = A_{w + 30}

where s = a small imaginary operator.

q = ( i_8 + i_32 + i_128 + i_512 + s )

Example 2:

What is the vector resultant of these two vectors?

( +i_2 ) * A_{90 degrees,0} + ( +i_32 ) * B_{45 degrees,-45}

| A | = 6

| B | = 10

+i_2  = +( -1 )^(1/2) which rotates a vector by

multiplication in a ccw direction pi/2 radians or 90

degrees.

+i_32 = +( -1 )^(1/32) which rotates a vector by

multiplication in a ccw pi/32 radians or 5.625

degrees.

The x axis and y axis are assumed the common

directrixs of the vectors A and B:

( i_2 ) * A_{90 degrees,0} = A_{90 + 90 degrees,0} =

A_{180 degrees,90 degrees}

( i_32 ) * B_{45 degrees,-45 degrees} =

B_{45 + 5.625 degrees,-45 + 5.625 degrees} =

B_{50.625 degrees,-39.375 degrees}

substituting | B | = 10 and | A | = 6

6_{180 degrees,90 degrees} + 10_{50.625 degrees,-39.375 degrees} =

7_{87.452,-2.548}

Example 3:

What is the limit of the following expression?

Limit [( -1 )^(1/(2^n))] * A_{c,d} = A_{c + pi/(2^n),d - pi/(2^n)}

n --> oo

As "n" becomes larger the value of "2^n" small

and "pi/(2^n) approaches 0.

Letting n --> oo then pi/(2^n) --> 0 then the

above becomes:

Limit [( -1 )^0] * A_{c,d} = A_{c + 0,d - 0} = A_{c,d}

n --> oo

Note: ( -1 )^0 = 1 therefor   1 * A_{c,d} = A_{c,d}

This suggest that the limit of "i_(2^n) = 1 as

n --> oo

Remember i_(2^n) = ( -1 )^(2^n)

Proof:

Let the complex number (-1,0) be square rooted as

many times as need to see the limit of the complex

number as the square roots continue.

first level    (0,1)

second level   (.70710678,.70710678)

third level    (.92387953,.38268343)

fourth level   (.98078528,.19509032)

fifth level    (.99518472,.09801714)

sixth level    (.99879546,.04906767)

seventh level  (.99969881,.02454122)

eighth level   (.99992470,.01227153)

ninth level    (.99998117,.00613855)

tenth level    (.99999529,.00306795)

. . .

thirtieth level (1,.00000000292583615853)

no rounding off of numbers, the imaginary part

and real part was truncated in the calculator

used to calculate the square root.

See the figure 3 on the opposing page for a

graphical representation of complex numbers as

they represent a vector.

Example 4:

What complex number rotates the vector D the same as

i_8?

i_8 * D = ( a + bi ) * D

what is "a" and what is b in the complex expression "a + bi"?

i_8 = ( -1 )^(1/8)

( a + bi) * D = a * D + bi * D

i_8 rotates the vector D 22.5 degrees or pi/8

radians.  Therefore a * D + bi * D must equal

D_{x + pi/8,y - pi/8}.  Since the magnitude of D is

not  changed when i_8 rotates the vector D.

( | a *  D | )^2 + ( | bi * D | )^2 = ( | D | )^2

after dividing the above expression by ( | D | )^2

( ( | a *  D | )^2 + ( | bi * D | )^2 = ( | D | )^2)

-----------------------------------------------------

( | D | )^2

a^2 + ( | bi | )^2 = 1  and tangent of angle of the complex

number (a , b) = tangent(pi/8 or 22.5 degrees).

tangent(pi/8) =.414218738744

b/a = .414218738744

b = .414218738744 * a

substituting .414218738744 * a for b

a^2  + ( .414218738744 * a )^2 = 1

solving for a:  a = .840894876315  ;  b = .348314415083

therefore ( .840894876315 + .348314415083i ) rotates the

vector D the same as "i_8" or ( -1 )^(1/8)

The convention of angle rotation in  a clockwise direction for

negative values is accepted with the following qualification:  All angle

rotations in a counter clockwise are negative as seen from a left-hand

viewing space; but, the same angle rotations as seen from right-hand

viewing space are positive.  What is the difference between RIGHT-HAND

VIEWING SPACE and LEFT-HAND VIEWING SPACE?  Refer to the opposing page

and figure 4.  A right-hand viewing perspective sees the directrix end at

the right of the initial point and the initial point on the left.  Where

the left-hand viewing space sees the opposite orientation for the same

vector or the initial point is seen at the right and the directrix is

seen on the left.  Therefore what rotates clockwise in a RIGHT-HAND

VIEWING SPACE  is rotating counter clockwise in a LEFT-HAND VIEWING

SPACE.  What is left directed in a right-hand viewing perspective is left

directed in a left-hand viewing space.  To have this graphically

demonstrated have an associate on the other side of a window draw a

vector from a figure in this book or rotate the hand with a marker in a

clockwise direction.  Then do the same from your side of the window and

note the apparent directions as you view the motion from the opposite

side of the window.

Notation in the treatise will be exclusively right-hand space

perspective denoted as follows:

Vector "A" in RIGHT-HAND VIEW SPACE is A_{x,y,z,...,w} or

| a |_{x,y,z,...W} the directrixs "x" "y" "z" "w" and others

will be at the right.

Vector "A" in LEFT-HAND VIEW SPACE is {w,...,z,y,x}_A or

{w,...,z,y,x}_| a | the directrixs "x" "y" "z" "w" and others

will be at the left in the curly brackets.

A vector in LEFT-HAND VIEWING SPACE is also a vector in RIGHT-HAND

VIEWING SPACE or in symbolic form:

{x}_A = A_{-x} or {-x}_| a + b | = | a + b |_{x}

Angles on the left of the vector magnitude denote left-hand viewing

perspective and angles on the right denote right-hand viewing

perspective or space.  What is seen as left in one space is seen as

right in the other space if the same vector is the subject of

observation.

Vectors represented graphically are as illustrated in figure 5 on

the opposing page for RIGHT-HAND VIEWING SPACE.

Vectors represented graphically are as illustrated in figure 6 on

the opposing page for the LEFT-HAND VIEWING SPACE.

Vectors can be reference using Cartesian, Polar, or other reference

systems.  No matter what the Reference System a vector is at minimum a

number pair.  One number specifies magnitude and the other number

specifies direction or a special relationship to its reference system.

Since there can be many reference systems for the same vector all

operations and functions on vectors may be either REFERENCE SYSTEM

INDEPENDENT OR REFERENCE SYSTEM DEPENDENT.  When an operation or

function of of a vector is reference system dependent or (rsd) it must

be so specified. Therefore operations and functions will be assumed as

reference system independent or (rsi).

Example 5 of a (rsi) vector operation:

let "a + bi" be vector defined by a complex number.

let "c + di" be a second vector defined by a complex

number.  n = an integer.

( a + bi ) + ( c + di ) = ( a + c ) + ( b + d )i

Rotate vectors "a + bi"; "c + di" by i_(2^n) or

i_(2^n) * ( a + bi ) = i_(2^n) * a + i_(2^n) * bi

i_(2^n) * ( c + di ) = i_(2^n) * c + i_(2^n) * di

Add the rotated vectors:

i_(2^n) * ( a + c ) + i_(2^n) * ( b + d )i

Now rotate the sum of the unrotated vectors or

complex numbers by (2^n).

i_(2^n) * [ ( a + c  ) + ( b + d )i ] =

i_(2^n) * ( a + c ) + i_(2^n) * ( b + d )i

Since the sum of the rotated vector is equal to the rotated sum of

the unrotated vectors, the operation of addition of complex number

represented vectors is REFERENCE SYSTEM INDEPENDENT or (rsi).

Example 6 of a (rsd) or reference dependent operation operation:

Let vector "A" be defined in terms of complex numbers or

a + bi = A.

Let vector "B" be defined in terms of complex numbers or

c + di = B.

Now rotate vectors "A" and "B" by "i_(2^n)" individually.

i_(2^n) * A = i_(2^n) * ( a + bi ) or

i_(2^n) * a + i_(2^n) * bi

i_(2^n) * B = i_(2^n) * ( c + di ) or

i_(2^n) * c + i_(2^n) * di

Now multiply the rotated results as follows:

[ i_(2^n) * A ] * [ i_(2^n) * B or

( i_(2^n) )^2 * ( ac + adi + bci + bd( i^2 ) )

Compare by multipling the unrotated vectors "A" and "B" or

A * B = ( ac + adi + bci + bd( i^2 ) )

Now rotate the results by "i_(2^n)" or

( i_(2^n) ) * A * B = ( i_(2^n) ) * ( ac + adi + bci + bd( i^2 )

Since the before and after rotations are not equal as below:

( i_(2^n) )^2 * ( ac + adi + bci + bd( i^2 ) ) =/=

( i_(2^n) ) * ( ac + adi + bci + bd( i^2 )

The operation of multiplication of vectors defined by complex

numbers is not REFERENCE SYSTEM INDEPENDENT or (rsi) but is

(rsd) or reference system dependent.  Dividing the above equation

by ( ac + adi + bci + bd( i^2 ) we have the following inequality

except for one case:

i_(2^n) =/= ( i_(2^n) )^2 or

1 =/= i_(2^n) if n --> OO per example 3

then

1 = i_(2^n)

When n --> OO the "2n"th level of "-1" is representable

by the complex number (1,0) = 1 and any number multiplied

by 1 is unchanged thus proving the above example.

Refer to the opposing page for numerical illustrations for four

cases where modulus and direction of a vector "B" is defined by

complex numbers.

Example 8:

Let "b" be the Modulus or absolute magnitude of the vector

"B" defined by defined by imaginary operators.

First Case: Where "B" is at 0 degrees or 0 radians.

and "b" equals 5.

| b |_{0,-90} or B_{0.-90}

B * B = B^2

if b = 5 then B^2 = | 25 |_{0,-90}

no change in direction indicated by

the multiplication.

Second Case: Where "B" is at 90 degrees or pi/2 radians.

and "b" equals 5.

| b |_{90,0} or B_{90.0}

B * B = B^2

if b = 5 then B^2 = | 25 |_{180,90}

A change in direction indicated by

the multiplication.

Third Case: Where "B" is at 180 degrees or pi radians.

and "b" equals 5.

| b |_{180,90} or B_{180.90}

B * B = B^2

if b = 5 then B^2 = | 25 |_{360,270}

A change in direction indicated by

the multiplication.

Forth Case: Where "B" is at 270 degrees or 3pi/2 radians.

and "b" equals 5.

| b |_{270,180} or B_{270.180}

B * B = B^2

if b = 5 then B^2 = | 25 |_{180,90}

Reverse change in direction indicated by

the multiplication.

If all of the above examples of the vector "B" were the same vector

but the reference system was rotated as seen from an independent vector

reference system, then this vector rotates depending upon the describing

reference system.  This means that the references to a vector must include

the reference to the reference system itself.  Yet the basic definition of

a vector is a number pair where one is the modulus or magnitude and the

other number signifies the direction.  To simplify Vector translation from

one reference system to another the operations upon the vectors must be

(rsi) functions and operational expressions.   If an absolute reference is

possible then all operations on the vector will be seen as the same.

Let us define a vector multiplication of vectors which is (rsi) such

as vector addition.  To do this we will normalize the vectors so the

multiplication is actually an addition.  This requires the concept of

POLARIZED EXPONENTIAL NOTATION OF A NUMBER/VECTOR or a POLARIZED

LOGARITHMIC NOTATION OF A NUMBER/VECTOR to be introduced.

Note: if a number "a" is raised to the power x and y then

if b = a^x and c = a^y it can be shown that

b * c = a^x * a^y = a^( x + y ) = b * c

^

|

--------------------------

| this expression reduces multiplication to an addition operation

which is the definition of normalization as referred to in the above

description of EXPONENTIAL NOTATION.  Now to define the meaning of

the word POLARIZED in the above description.

Let "a" be a number and let "E" be a vector then " a^E " is a

Polarized number such that

a^E = | a^e |_[ E/e ] = | a^e |_{direction of "E" }

Which means the vector with modulus "a^e" and has the same direction

as "E".  The notation " [ some vector ] "  refers to the array of

vectors which result in some vector.  Therefore if  vectors below

were to use this notation then

| 34 |_{90} = B

| 23 |_[ B/b ] = | 23 |_{90} = | 23 |_[ (| 34 |_{90})/34 ] =

| 23 |_[ | 1 |_{90} ] = | 23 |_{90}

Also it will be here set as  convention to represent an array

of vector after the following manner:

[ X Y Z ] = | absolute value of "X + Y + Z" |_{x,y,z} =

| absolute value of "X + Y + Z" |_[ ( X + Y + Z )/(| X + Y + Z |) ]

A numerical example can be illustrated as follows:

[ 1 2 34 ] = vector  1 along x axis

vector  2 along y axis

vector 34 along z axis

[ X  Y  Z ]

^  ^  ^

|  |  |__ z axis magnitude of 34.

|  |_____ y axis magnitude of  2.

|________ x axis magnitude of  1.

[ 1 2 34 ] = | 34.073 |_{88.318 deg,86.635 deg,3.7627 deg}

[ 1 2 34 ] = | 34.073 |_[ .0293 .0587 .9978 ]

.0293 .0587 .9978  are the cosines of the x,y,z angles of

{88.318 deg,86.635 deg,3.7627 deg} of {x,y,z} directrixs.

{x,y,z} =/= [ X/x Y/y Z/z ]

as

{x,y,z} is an array of angles

[ X/x Y/yZ/z ] is an array of vectors

{ x,    y,   z }

^     ^    ^

|     |    |______ angle from z axis toward the x axis

|     |___________ angle from y axis toward the z axis

|_________________ angle from x axis toward the y axis

See the opposing page for the graphic representation of the

above convention.

The power in mathematics is in its notation; and the difficulty

in mathematics is in its notation.  The digression just taken is to

simplify the definition of vector multiplication through normalization

by exponentiation and logarithms.  Too facilitate the symbolic notation

by the standard keyboard characters, the following notation will also

be followed:

e^n       = exp[e](n)

ln n      = log  base "e" of "n"

log n = log  base "10" of "n"

log[e] n  = ln n

alog[e] n = e^n

aln n     = alog[e] n = exp[e](n) = e^n

Now let us define vector (rsi) multiplication through normalization

by exponentiation and logarithmics operations.

Given two vectors "A" and "B" such that

A * B =

| exp[r]( | log[r](| A |) |_[ A/a ] + | log[r](| B |) |_[ B/b ] ) |_

[    | log[r](| A |) |_[ A/a ] + | log[r](| B |) |_[ B/b ] /

| | log[r](| A |) |_[ A/a ] + | log[r](| B |) |_[ B/b ] |  ]

Where "A" and "B" may be any vector with "n" directrixs or components:

A = [ W . . . X Y Z ] or | a |_{w,...x,y,z}

B = [ W . . . X Y Z ] or | b |_{w,...x,y,z}

Example 9:

Find the (rsi) product of the following vectors.

1.  4_{45,45} * 8_{45,45} = P

per the above angle measurement.

normalizing the vectors by selection base or radix 2

log 4 = 2

log 8 = 3

The product becomes the addition of following vectors and

its exponentiation of the resultant.

2_{45,45} + 3_{45,45} = | 5 |_{45,45}

P = | 2^5 |_{45,45} = 32_{45,45}

2. 23_[ X Y Z ] * 55_[ X1 Y1 Z1 ] = P

What is the product value of "P" ?

where X = cosine of angle from x axis

Y = cosine of angle from y axis

Z = cosine of angle from z axis

X1 = cosine of angle from x axis

Y1 = cosine of angle from y axis

Z1 = cosine of angle from z axis

or direction cosines of vectors.

Let X  = cos(44)  for the 23 magnitude vector in degrees.

Let Y  = cos(14)  for the 23 magnitude vector in degrees.

Let Z  = cos(49)  for the 23 magnitude vector in degrees.

Let X1 = cos(144) for the 55 magnitude vector in degrees.

Let Y1 = cos(14)  for the 55 magnitude vector in degrees.

Let Z1 = cos(149) for the 55 magnitude vector in degrees.

Normalizing the two vectors using base "e" or natural logarithms.

ln(23_[ cos(44) cos(14) cos(49) ]) = | ln23 |_[ cos(44) cos(14) cos(49) ]

ln(55_[ cos(144) cos(14) cos(149) ]) = | ln23 |_[ cos(144) cos(14) cos(149) ]

The intermediate product becomes the addition of the normalized

vectors and subsequent exponentiation.

ln 23 = 3.13549421593

ln 55 = 4.00733318523

unit vector for 23 magnitude vector =

[ .719339800339 .970295726276 .656059028991 / magnitude of the

sum of X Y Z vectors ]

unit vector for 55 magnitude vector =

[ -.809016994375 .970295726276 -.857167300702 / magnitude of the

sum of X1 Y1 Z1 vectors ]

Once the unit vectors are found the logarithmics sum is taken:

3.13549421593_[ .52333451203 .705910113963 .477296448268 ] +

4.00733318523_[ -.529923003767 .635563936711 -.56146245861 ] =

4.84365385369_[ -.099649090899 .982789785971 -.155545155079 ]

exponentiating the above sum the product becomes:

e^( 4.84365385369_[ -.099649090899 .982789785971 -.155545155079 ] )

or

126.932297491_[ -.099649090899 .982789785971 -.155545155079 ]

using the same parameter of input as output the answer is

126.932297491_[ cos(95.7189639361) cos(-10.6452457619) cos(81.0515851808) ]

3. Using the Vectors in Example 9.2 the perspective independent

version of angular parametered vectors is as follows:

( Note before hand that vector directed or unit vector directed

vectors are perspective independent and do not rely on a

perspective to determine which direction the angles rotate for

positive or negative rotation, such as counter clockwise is

see as a positive angle.  Below the example is that all angles

are measured toward a neighboring axis are positive regardless of

the perspective. )

23_[ cos(44) cos(14) cos(49) ]) =

[ 12.0366937767 16.2359326211 10.9778183102 ] or

23_{58.4438090541,45.0968822492,61.4910228451}

^             ^             ^

|             |             |

|             |              -- likewise z axis to x axis

|              -- angle y axis toward z axis

-- angle measured from x axis toward y axis

55_[ cos(144) cos(14) cos(149) ]) =

[ -29.1457652072 34.9560165191 -30.8804352236 ] or

55_{122.000252651,-50.5381768421,124.156996953}

^             ^             ^

|             |             |

|             |              -- likewise z axis to x axis

|              -- angle y axis toward z axis

-- angle measured from x axis toward y axis

23_{58.4438090541,45.0968822492,61.4910228451} *

55_{122.000252651,-50.5381768421,124.156996953} =

126.932297491_[ cos(95.7189639361) cos(-10.6452457619) cos(81.0515851808) ]

or

126.932297491_{95.7189639365,-10.6452434747,98.948414819}

The above answer is both perspective independent and

reference system independent or (p-i) and (rsi).

(note the hyphen in p-i is to distinguish the acronym from

"pi" the angular measurement in radians or half the

circumference of the circle)

4. Compare the (rsi) product of the following complex numbers

with the (rsd) product of the complex numbers?

( 3 + 45i ) multiplied ( 33 - 3i )

(rsd)

( 3 + 45i ) X ( 33 - 3i ) = ( 234 + 1476 )

absolute = 1494.4333672 angle = 80.991496258 deg

(rsi)

( 3 + 45i ) * ( 33 - 3i) = ( 121.312143295 + 113.004404959i )

absolute value = 165.790927529 angle = 42.9694176819 deg

ANGLE BETWEEN TWO VECTORS CAN BE FOUND WITH THE DOT PRODUCT

The DOT product of two vectors results in a number which represents

the product of the magnitude of the two vector times the cosine

of the angle between the vectors.  Thus

A . B =  a  *  b * cosine(angle between "A" and "B")

also if

A = | a |_[ X/a   Y/a   Z/a ]   where [ X  Y  Z ] = A

B = | b |_[ X1/b Y1/b Z1/b ] where [ X1 Y1 Z1 ] = B

A . B = ab( (x/a)(x1/b) + (y/a)(y1/b) + (z/a)(z1/b))

(by definition of DOT product)

dividing both side by "ab" we have

cos(A to B) = (x/a)(x1/b) + (y/a)(y1/b) + (z/a)(z1/b)

or

cos(A to B) = cos(X to A) * cos(X1 to B) +

cos(Y to A) * cos(Y1 to B) +

cos(Z to A) * cos(Z1 to B)

see figure on opposing page for illustration

A PUZZLE FROM LEONARD EULER CONCERNING "e" AND THE FORMULA

NAMED AFTER HIM:

e^[(r)i] = cos(r) + (i)sin(r)

The above formula is known as Eulers formula in most math

text books.  It was derive by taylors series approximations of "e"

"cos" and "sin".  No reproof or iterations will be done here as it is

beyond the scope of this treatise.  Yet a salient point concerning the

effect of notation upon mathematical conclusions can be learned here.

The (rsi) multiplication highlights an interesting property on the

effects of notation.  The product of to vectors represent in eulers form

are as follows:

e^[(r)i] * e^[(s)i] = (cos(r) + (i)sin(r)) * (cos(s) + (i)sin(s))

Now the exponential side of the equation can be written in two forms

as next follows:

e^[(r)i] * e^[(s)i] = e^[(r + s)i]  or  e^[((r + s) / 2)i] * 2

^                  ^

|                  |

--------------------------                  |

| (rsd) reference dependent multiplication   |

|

|

---------------------------------------------

| (rsi) reference independent multiplication

The choice of multiplying and dividing by 2 results in "1" yet one

form of vector notational treatment ADDS the arguments or angles and the

other notational form AVERAGES the arguments or angles.  Please note

that:

e^[(r)i]  =  (cos(r) + (i)sin(r)

e^[(s)i]  =  (cos(s) + (i)sin(s)

e^[(r)i] + e^[(s)i] = (cos(r) + (i)sin(r) +  (cos(s) + (i)sin(s)

e^[(r)i] + e^[(s)i] = (cos(r) + cos(s) + (i)sin(r) + (i)sin(s)

With the trig identities of

cos(x) + cos(y) =  2cos((x + y) / 2)cos((x + y) / 2)

sin(x) + sin(y) =  2sin((x + y) / 2)cos((x + y) / 2)

The right hand side of the equation becomes:

2cos((r - s) / 2)(cos((r + s) / 2) + (i)sin((r + s) / 2)

Now  the equation becomes:

e^[(r)i] +e^[(s)i] = 2cos((r - s) / 2) * [cos((r + s) / 2) +

(i)sin((r + s) / 2)]

Note: e^[((r + s) / 2)] = [cos((r + s) / 2) + (i)sin((r + s) / 2)]

e^[(r)i] + e^[(s)i] = 2cos((r - s) /2) * e^[((r + s) / 2)]

or

e^[(r)i] + e^[(s)i] = 2cos((r - s) /2) * e^[((r + s) / 2)]

e^[((r + s) / 2)] = ( e^[(r)i] + e^[(s)i] ) / 2cos((r - s) / 2)

Therefore  the (rsi) form of Eulers form of vector multiplication is

is as follows:

e^[(r)i] * e^[(s)i] =

( [cos((r + s) / 2) + (i)sin((r + s) / 2)] / 2cos((r - s) / 2) )^2

As:

e^[(r)i] * e^[(s)i] = e^[(r)i + (s)i] =

e^[(r + s)i]  or  e^[((r + s) / 2)i] * 2

^                  ^

(rsd)              (rsi)

****************************************************************

Axiom 1: Any vector added to itself results in a vector in the same

direction as itself and magnitude twice the original vector.

Axiom 2: Any vector multiplied by itself results in a vector

in the same direction as itself with a magnitude according to the

square of the root magnitude in any 'reference system independent'

multiplication. Therefore for (rsi) multiplication:

( [cos((r + s) / 2) + (i)sin((r + s) / 2)] / 2cos((r - s) / 2) )^2

does not rotate when squared.

****************************************************************

LET US REVIEW TRADITIONAL VECTOR ANALYSIS IN VIEW OF THE

AFORE GOING (rsi) DEVELOPMENT OF VECTOR MATHEMATICS

VECTOR PROPERTIES:

1. A = A                        This property states equality is

tautological and reflexive.

2. if A = B then B = A          This property states Identity is

symmetric to both sides of the

equation.

3. if A = B and B = C then      This property states substitution

A = C                        of identities are transitive

4. if A = B and C = D then      This property states substitution

A + C = B + D                is applicable to vector operations

A * C = B * D

A . C = B . D

A X C = B X D

5. A + B = C                    This property states vector

A * B = C                    addition or multiplication has

A X B = C                    a closure properties such that

A . B = c    *(number)       two vectors under and operation

can be represented by single

vector and visa versa

* ( The dot product is exempt of

this property as the resultant

"operation" results in a number

which is a "vector" with always

and only one direction.)

6. (A + B) + C = A + (B + C)    Associative property of Vector

(A * B) * C = A * (B * C)    operations work only with the

(A . B) . C  = ?             addition and (rsi) multiplication

(A X B) X C  =/= A X (B X C) The dot product is undefined and

the cross product fails.

CROSS PRODUCT OF A,B VECTORS = A X B =

a * b * sin(angle A to B) * [ unit vector 90 degrees A,B plane ]

CROSS PRODUCT OF B,A VECTORS = B X A =

b * a * sin(angle B to A) * [ unit vector 90 degrees A,B plane ]

Since sin(angle B to A) = -sin(angle A to B)

Therefore:

A X B = -1 * ( A X B )

7. A + B = B + A                This property states that vector

A * B = B * A                operations are commutative except

A . B = B . A                for the cross product which is

A X B = -(B X A)             equal to minus the commuted product.

(that is rotated 180 degrees)

8. A + (-1)A = 0                Two vectors with equal absolute

A X A = 0                    magnitude but with opposite

direction added define zero.

(cross product of equal vectors

defines zero)

9. A + 0 = A                    Zero or Null vector property for

A * 0 = 0                    vector operations defined.

A . 0 = 0                    A Null vector is a vector of zero

A X 0 = 0                    magnitude and a real argument of

direction | 0 |_{x,y,...,z}

10. A . (B + C) = A . B + A . C  Distributive property applies for

A X (B + C) = A X B + A X C  dot and cross product only and

A * (B + C) =/= A * (B + C)  fails for the (rsi) multiplication

or normalization form of

multiplication.

11. m(A) = | ma |_[ A/a ]        scalar multiplication increases

magnitude and angular moment when

m < 0 or imaginary.

DEFINITIONS BASED UPON THE PRECEEDING AXIOMS AND PROPERTIES:

1. A vector is at least a number pair of magnitude and direction.

2. The vector magnitude is always greater than 0.

3. Any vector where the magnitude is less than 0 or has imaginary

component has implicit direction.

4. A null vector is a vector with 0 magnitude and one direction.

5. Zero is a number if no direction specified.

6. The sum of two null vectors result in a null vector of angle

equal to the sum of the two angles divided by two.

7. Vector division results in a vector.

8. A vector can be represented by either a vector product or a

vector sum or both.

9. The absolute value of a vector is greater than zero regardless

of any implicit direction to the angular direction.

10. The absolute value of a vector magnitude is a positive number

ALWAYS or is equal to the magnitude of the absolute value of

the vector.

The absolute value of the vector is represent by the

following notation:

abs(A)   = absolute value of vector "A"

removes the direction of "A" and leaves the positive

magnitude only.

A / abs(A) = unit direction vector of | 1 |_[ A / abs(A) ]

11. A pure vector is a vector with no implicit direction in the

magnitude.

We will use the functional notation to represent the

different factored values from the vector parameters

pur(A)   = removes the implicit direction in the

magnitude and places the implicit direction

in the argument or angular parameter of "A"

pur(A) is a pure vector but may not have the same angular

argument as "A"

pur(A) =/= A

(if the magnitude of "A" has implicit direction)

mag(A) removes the direction from "A" and leaves the

magnitude only even if it has implicit direction.

mag(A) = a

arg(A) removes the magnitude from A and equals the direction

parameters or angles only.

if A = | 23 |_{ 356 degrees } then arg(A) = 356 degrees

if A = | 24 |_[ (any unit vector) ] then

arg(A) = [ (any unit vector) ] or the angle of the

unit vector.

| mag(A) |_{ arg(A) } = A = | a |_[ A / abs(A) ]

VECTOR ADDITION TO MULTIPLICATION CONVERSION AND

VECTOR MULTIPLICATION TO ADDITION CONVERSION.

Since multiplication of two vectors produces a vector and

the addition of two vectors results in a vector,  conversion from

addition to multiplication and visa versa is possible.

let A + B = C and D * E = P and P = C then

A + B = D * E  where arg(A) = arg(D)

arg(B) = arg(E)

Find the Magnitude of "A", "B", "D", "E" such that

A + B = D * E

Refer to figure () on the  opposing page.

P = e^( | ln(abs(D)) |_[ D / abs(D) ] + | ln(abs(E) |_[ E / abs(E) ])

or

ln (P) =  | ln(abs(D)) |_[ D / abs(D) ] + | ln(abs(E) |_[ E / abs(E) ]

Per the figure across the similar parallelograms are formed therefore

all sides of the vector differ by the ratio of:

abs(P) / abs(ln (P)) = r

or

abs(e^( | ln(abs(D)) |_[ D / abs(D) ] + | ln(abs(E) |_[ E / abs(E) ]))

----------------------------------------------------------------------- = r

abs(| ln(abs(D)) |_[ D / abs(D) ] + | ln(abs(E) |_[ E / abs(E) ]))

e^( | ln(abs(D)) |_[ D / abs(D) ] + | ln(abs(E) |_[ E / abs(E) ]) = D * E

| ln(abs(D)) |_[ D / abs(D) ] + | ln(abs(E) |_[ E / abs(E) ] = ln(D) + ln(E)

D * E = r ( ln(D) + ln(E) )

D * E = r(ln(D)) + r(ln(E)) = P = C

Since:

[ D / abs(D) ] = [ A / abs(A) ]

[ E / abs(E) ] = [ B / abs(B) ]

then:  A = r(ln(D)) and B = r(ln(E))

substituting below renders the following equations which was to

be found.

D * E = A + B = C = P

*************************

To convert an addition A + B to product D * E  such that:

e^(ln(abs(A + B)) / abs(A + B))A * e^(ln(abs(A + B)) / abs(A + B))B =

A + B = D * E

set D = e^(ln(abs(A + B)) / abs(A + B))A

set E = e^(ln(abs(A + B)) / abs(A + B))B

*************************

To convert a product A * B to addition D + E such that:

(abs(e^(lnA + lnB)) / abs(lnA + lnB))( lnA + lnB ) = A * B = P = D + E

set D = (abs(e^(lnA + lnB)) / abs(lnA + lnB))( lnA )

set E = (abs(e^(lnA + lnB)) / abs(lnA + lnB))( lnB )

POINTS IN SPACE

Two vectors described by the logical OR statement can define two

points in space.

A (OR) B = points (A,B)  if A = B then Point (A) (AND) B

Two vectors describe by the logical AND statement can define only

one point in space.

A (AND) B = point (C) C is a unique point

A + B, A * B are specific applications of the AND function if

A,B are arbitrary constants.

X + Y, X * Y are specific applications of the OR function where

"X" and "Y" are variables defining only one of a se are selectable.

Therefore Vectors (OR)'ed describe a set;

Vectors (AND)'ed describe a unique vector.

LINES IN SPACE

Given a line in cartesian form or the general line:

ax + by = c

let | x |_{ 0 }  = x-axis

let | y |_{ 90 } = y-axis

Solving the line equation for "y" or "f(x)".

f(x) = y = (c -ax) / b

therefore a line in vector form is

| x |_{ 0 } + | f(x) |_{ 90 } = set of vectors C

see figure () across page.

Lines  can also be represented by the vector constant "A" plus

a vector representing the slope of the line or | f(s) |_{ slope angle }.

Where "A" is the Distances from the initial point to the line.  "S" is

the direction of the slope of the vector per the figure across from the

page.

| f(s) |_{ q } where q = slope angle

| a |_{ p }    where p = perpendicular angle to q

q = p + 90 degrees

q = arctan( a/b ) where the line is

describe by the cartesian line ax + by = c.

(s) is a variable that spans the length of the line.

a   is a constant or the closest distance to line from

the initial point.

Lines can also be expressed as vectors with its magnitude as a function

of the angel argument.

| f(q) |_{ q } where the q is in radians.

example:

given the line ax + by = c in cartesian form

and converting to the polar form by setting

x = r * cos(q) y = r * sin(q)

then line ax + by = c can be express as below

r * ( a * cos(q) + b * sin(q) ) = c

or

r = c / ( a * cos(q) + b * sin(q) )

Therefore the vector form of the line in terms of (q) is per figure

across the page.  And expressed below:

line (q) = | ( c / ( a * cos(q) + b * sin(q) ) |_{ q }

where line(q) = vector R

for all values of q such that

a * cos(q) + b * sin(q) =/= 0

CIRCLES IN SPACE

The cartesian form of a circle is:

x^2 + y^2 = h^2   where h is the radius of the circle.

if

x = r * cos(q) y = r * sin(q)

converting to polar form the equation becomes:

( r * cos(q) )^2 + ( r * sin(q) )^2 = h^2

solving for r in the above equation

r = ( h^2 / (( cos(q) )^2 + (sin(q))^2 ))^(1/2)

since ( cos(q) )^2 + (sin(q))^2 ) = 1

r = (h^2 / 1)^(1/2)  and r = h

The general form of the vector with center at the initial point is:

| h |_{ q }

The general form of the vector with center at "A" distance from the

initial point is:

| h |_{ q } + A or | h |_{ q } + | a |_ { arg(A) }

ELLIPSE IN SPACE

The cartesian form of a ellipse is:

(x^2)/(a^2) + (y^2)/(b^2) = 1

if

x = r * cos(q) y = r * sin(q)

converting to polar form the equation becomes:

( r * cos(q) )^2 / (a^2) + ( r * sin(q) )^2 / (b^2) = 1

solving for r in the above equation

r^2  = ( 1 / (( cos(q) )^2 / (a^2) + (sin(q))^2 / (b^2) ))^(1/2)

r = ( (a * b) / ( a * (cos(q))^2 + b * (sin(q))^2 ) )^(1/2)

The general form of the ellipse centered at initial point is therefore:

| ( (a * b) / ( a * (cos(q))^2 + b * (sin(q))^2 ) )^(1/2) |_{ q }

The general form of the ellipse center "A" distance from the initial

point is therefore:

| ( (a * b) / ( a * (cos(q))^2 + b * (sin(q))^2 ) )^(1/2) |_{ q }

+ | a |_{ arg(A) } = ellipse "A" distance from initial point.

PARABOLA IN VECTOR SPACE

The Cartesian form for the Parabola opening up the y-axis is

y^2 = 4 * a * x

Converting to polar for to prepare for vector equivalent

where:  y = r * sin(q)  and x = r * cos(q)

(r * sin(q) )^2 = 4 * a * r cos(q)

r = 4 * a * cos(q) / (sin(q))^2

Therefore a parabola in vector form is

| 4 * a * cos(q) / (sin(q))^2 |_{ q }

Note: "q" cannot be pi/2 radians or magnitude of the

vector must be infinite.  Therefore the parabola opens

up at the "0" degrees or "0" radians.

note: add any angle "h" to "q" to rotate the parabola.

| 4 * a * cos(q) / (sin(q))^2 |_{ q + h }

Also we can rotate vectors by multipling by rfi vector

multiplication therefore the parabola can be rotated as

below method shows:

(i_2n + i_2t + ...) * | 4 * a * cos(q) / (sin(q))^2 |_{ q }

Adding a vector "M" to the parabola translates its location

the magnitude and direction of the vector "M" rotated "h" or

i_2n + i_2t + ... .

M + | 4 * a * cos(q) / (sin(q))^2 |_{ q + h }

M + (i_2n + i_2t + ...) * | 4 * a * cos(q) / (sin(q))^2 |_{ q }

Solving the equation (r * sin(q) )^2 = 4 * a * r cos(q) for q or:

q = Arccos(-4a +/- ((4a)^2 + 4 * r^2)^(1/2) / 2r)

The Parabola can be represented in terms of r as follows:

| r |_{  Arccos(-4a +/- ((4a)^2 + 4 * r^2)^(1/2) / 2r) }

rotated by "h" and translated by "M" as below:

M + | r |_{  Arccos(-4a +/- ((4a)^2 + 4 * r^2)^(1/2) / 2r) + h } ```