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
radians.
( -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.
Add the vectors:
( 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[10] 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[2] 4 = 2
log[2] 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 }
