, , . , ,
: , , , ,
-----------------------------------
, , . , ,
: , , , ,
-----------------------------------
An algorythm of presentation of vector algebra multiplicative compositions in form of qua-ternionic matrices are shown. New identities of vector algebra, as well as old one, are found. New formulas of matrix presentation of vector algebra operations are found
Key words: quaternionic matrices, vectorial matrices, multiplicative compositions, identities of vector algebra, matrix presentation ------------------ -----------------------
512.643.8:531.3

..
, -

. , 8, . , , 49005 .: 067-726-07-72; (056) 748-07-06
..

ї* .: 050-321-14-60
..

ї* .: 067-921-10-67; (056) 713-58-03 * .

. , 2, . , , 49010

. , [19, 26, 28]. [18, 27, 29]. , [7, 23]. , , , , , , ..
[1, 17]. - [3, 5, 25], [10, 11, 24], [21, 22], [9, 12, 13, 31, 32, 33], [30]. , , .

- (4 4), Ei E^ ( = 1,2,3) [14, 15]:
3
0 1 2 0 1 2
1 0 2 , 0 = 1 0 2
2 0 1 2 0 1
2 1 0 2 1 0
=
- (4 1):
0
.
- (4 1):
44 + 22 + 33


23 - 32
1 - 1 2 2
, , , . .

I. ,,... 0,0,0^0,...,
- 0,0,0^0,...
II. :

/-


-


\
-
\ 0
>1





-

\

^
-

1
0\

III. :
1 0 ' 0 + 0 ' 0 =(0 + 0 ) 0 ;
2 0 ' 0 ' 0 + 0 ' 0 ' 0 + 0 ' 0 ' 0 + 0 ' 0 ' 0 = .
(0 + 0 )(0 + 0 )
0 0 0 ^ + 0 0 0 d + 0 0 0 d + + 0 0 0 ^+0 0 0 ^+0 0 0 ^ + +0 0 0 d+0 0 0 ^ =
= (0 + 0 )(0 + 0 )( + 0 )d0;
..
IV. , :
1. _ , ( ):
0 ' 0 , 0 ' 0,
2. ,, ( ):
0 0 0 0 < II 0 0 ∞
0 0 0 0 (0 0 ) = (0 0 ∞
> 0 0 = 0 ? II 0 0 ∞
0 0 0 = 0 (?0 II 0 0 ∞
3. ,,^ ( ):
0'0' 'do = 0 {0 '(0'do)] = 0 '[(0'0)'^] =
[(0 ' 0 ) ' ]' do = [ 0' (0 ' 0 )] ' do = (0 ' 0 ) ' (0 ' do ),
.
V. , , . :
1. :
:
1. , , .. 0 -0 0 -0 ( : 21 );
2. ,,, .. 0'0'0, 0'0'0, 0 '0'0, 0' 0'0, ( : 22 );
3. ,,^, .. 0 0 0'd0,
0 '0 '0' d0, 0 ' 0' 0 ' |, 0 '0 '0' |, 0 '0 '0 ' |, 0 '0 '0'd0, 0'0'0'd0, 0'0'0'd0, ( : 23) ..

21
, - .
-
, 0 0 ^

2. : 0 (0 0 )^
0 ^(0 0 )^
0 (0 0 )^
0 (? 0 )^
( )
( )
( )+ ( )
) ^
( ) ( )
( )
( ) ( )
X )
( )+ ( )
=
0

2

3

(0 ' 0 )' 0 ^ (0 ' 0)' 0 ^ (0'0)'0 ^ (0' 0)' 0 ^
( )'
- (' ) + ( ) -( )'
(' ) - 2(' ) + ( )
( )' (' ) - 2 (' ) + ( )
-( )' - (' ) + ( )
3.
. :
0 [0 (0 do)]
, [(, , )(1, ]
[(0 0)]do
-(')('d)+ [( d)]
- [ ( d)]-(a )(' d)+ [ ( d)]
-(')( ') + 1'[( )(1]
-I [( ()' ]-(" )(' () + [( ) (1 ]
- (' ) (' d) + [( ) ]' d
, , :
(0 - 0 )(0 - 0 )0 =
= 0 '(0 ' 0 )- 0 '(0' 0 )-0 '(0 ' 0 ) + 0 '(0' 0 ),
(0 + 0 )(0 - 0 )0 =
= 0 '(0 ' 0 )- 0 '(0' 0 ) + 0 '(0 ' 0 )- 0 '(0' 0 ),
(0 - 0 )(0 + 0 )0 =
= 0 '(0 ' 0 ) + 0 '(0' 0 )-0 '(0 ' 0 )- 0 '(0' 0 ),
(0 + 0 )(0 + 0 )0 =
= 0 '(0 ' 0 ) + 0 '(0' 0 ) + 0 '(0 ' 0 ) + 0 '(0' 0 ).
(0 - 0 )(0 - 0)0 =
= (0 ' 0 )' 0 -(0 ' 0)' 0 -(0 '0 )' 0 +(0' 0)' 0, (0 + 0 )(0 - 0)0 =
= (0 ' 0 )' 0 -(0 ' 0)' 0 +(0 '0 )' 0 -(0' 0)' 0, (0 - 0 )(0 + 0)0 =
[0 (0 )] do
-[( )' ] d -(' )( d) + [(a ) ] d - (' ) (' d) + [ ( )]' d
(0 0 )(0 do )^
[ ( )]d- (' )( d) + [ ( )]d - (' ) (' d) + ( )' ( d)
-( )( ' d)-(a' )( d) + (a )( d)
VI. :
1. :
(0 + 0) 0 = 00 + 00;
(0 - 0) 0 = 00 - 00;
2. :
(0 - 0 )(0 - 0) 0 =
= 0 ' 0 ' 0 - 0 ' 0' 0 - 0' 0 ' 0 + 0' 0' 0,
(0 + 0 )(0 - 0) 0 =
= ' ' - 3' + 3' ' - 3'3'
0 0 0 0 0^0 0 0 0 0 0>
(0 - 0 )(0 + 0) 0 =
= ' + 3' - 3' ' - 3' 3'
0 0 0^0 -0 0 0 0 0 0 0 0
(0 + 0 )(0 + 0) 0 =
= (0 ' 0 )' 0 +(0 ' 0)' 0 -(0 '0 )' 0 -(0' 0)' 0, (0 + 0 )(0 + 0)0 =
= (0' 0)' 0 +(0' 0)' 0 +(0 '0)' 0 +(0' 0)' 0.
-
:
(0 + 0 )(0 + 0 )( + 0 )do =
= AoBoCodo + AoB0Codo + 000^ + A0B0Codo + +AoBoC0do + + A0BoC0do + 000^,
(0 - 0 )(0+0 )(0+0) ^ =
= AoBoCodo + 000^ - A0BoCodo - 000^ + +AoBoC0do + AoB0C0do - ^, - A0B0C0do,
(0 + 0 )(0 - 0 )( + 0) do =
= AoBoCodo - AoB0Codo + 000^ - A0B0Codo + +00 - AoB0C0do + A0BoC0do - 00,
(0 + 0 )(0 + 0 )(0 - 0) do =
= 000^ + AoB0Codo + A0BoCodo + 000^ --AoBoC0do - - A0BoC0do - A0B0C0do,
(0 - 0 )(0 - 0 )(0+0) ^ =
= AoBoCodo - 000^ - A0BoCodo + A0B0Codo + +00 - AoB0C0do - + A0B0C0do,
= ' ' + '3' + 3' ' + 3 - 3 -
0 0 0^0 0 0^0 0 0^0 0 0*
(0 - 0 )(0 + 0 )(0 - 0)do =
= 000^ + AoB0Codo - 000^ - A0B0Codo -
-AoBoC0do - + A0BoC0do + 00,

3
(0 + 0 )(0 0 )(0 0) ^ =
= ABCd AB0Cd + A0BCd A0B0Cd
ABC0d + AB0C0d + A0B0C0d,
( 0 )( }(0 ) ^ =
= ^ AB0Cd 0^ + A0B0Cd
^ + AB0C0d + A0BC0d ,
, ( ), . :
( + 0 )( + }( + } ^ =
= [ (}]+ [ (}]+
+0 [ (}]+0 [ (Cd}]+
+ [ (0d}] + [ (}] +
+0 [ (}]+0 [ (}],

(0 + 0 )(0 + 0 )(0 + 0) do =
= 0 [(00 )do ] + 0 [(00 )do ] +
+0 [(00 )do ] + 0 [(0) do ] +
+0 [(00 )do ]+0 [(00) ^ ]+
+0 [(00 ^0 ] + 0 [(00) do ],
VII. :
1. :
11 + 0 ^ 2
..
11 ( + 0 } ^ 2
1.2. ( 0} ^ 2
[


2. :
2.0.1. 0 (00) + 0 (00) + 0 (00) + 0 (00) ^ 4
2.0.2. (} + (0} 0 (} 0 (0 } ^ 4
( }
2.0.3. (} 0 (0} + 0 (} 0 (0} ^ 4
20.4. (} 0 (0} 0 (} + 0 (0 } ^ 4
(} + (0} + 0 (} +
1.1.1.
+0 (0 }^4
( } ( }
. (} + (0} 0 (} 0 (0 } ^ 4 (} (0} + 0 (}
1.1..
0 (0 }^4
( }-
(} (0} 0 (} +

1.1.4.
.. +0 (0 }^ 4
( } ( } ( + 0}( + 0} ^4
1.1.
( }
=4
( } ( }
1.2. ( 0}( + 0} ^4
1.3. ( + 0}( 0} ^4
1.4.
( 0}( 0}0 ^ 4
0 = 4 0

0 0
( }
= 4
( )

( }
=4
( } ( )
, [8]:
( ) = ( } ( }, ( } = ( } ( }, ( } = ( ^ .
2. , :
0 [0 (Codo)] + 0 [0 (Codo)] + 0 [0 (Codo)] + 311. + 0 [0 ()]+0 [0 ()]+0 [0 (c0do)] -
+0 [0 (C0do)] + 0 [0 (C0do )]^ 8
..
, - :
(}(d)
0










(0 + 0 )(0 + 0 )(Co + C0)^ ^ 8
[( ) d] - (' ) (' d)
-(')('d) [(d)]
0 0
= 8
=8
0
= 0
), [ ( d)]
0
( )( d)
4 (0+0 )(0 - 0 )(Co - co) ^,
^ 8 (0- 0 )(0- 0 )(0 - c0) do,
..
. :
[ ( d)] = [ ( d)] = ( )'( d) = ( )( d) =
' ' d ' ' d d ' ' d ' ' d ' ' d
( d) ( d)
[( )' d] -[( )' d] +
+ [( )' d] - [( )' ] d = 0,
[( )' d] = (' )( d)- (' )( d) + (' d)(a ), [ ( d)] = (a )( d), .
, [2, 8, 20].
VIII. , . :
1.
'
- (0 + 0 ) 0,
0

1
- (0 - 0 ) 0
2. ( )
^ 4 (0 + 0)(0 - 0)0,
( )
1
-( 0 - 0 )(0 - 0)0
1 [(0 + 0 )(0 + 0 )-(0 + 0 )(0 + 0 )](0 - 0 )do
3.

- , . , , , .
, , , , , , , .. [4].

1. . . / . .
- .: , 1969. - 368 .
2. .. . / .. , .. - : ї, 1978. - 216.
3. .. . / .. , ..
- .: , 1973. - 320.
4. .. / .. // . - 1980. - 2. - .3-13.
5. .. , , / .. // . - 1970.
- 8. 1. - . 13-19.
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- 544.
7. .. . / .. - .: - , 1995. - 240.
8. ., . . - .: , 1984.
- 832.
0
..

0
0
0
9. .. . . / ..
- .: , 1985. - 88.
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11. .. . / .. // . - 001. - 1. - .148-157.
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. .. . . / .. - : . , 1983. - 08 .
33. .. : . / .. - .: , 00. - 51.
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. .. - . / .. , .. - .: , 1973. - 60.
38. . . / , . , : . . - .: , 1980. - 80.
39. . . / , . , .
- .: . . ., 1950. - 445.
30. . . / . , . - .: , 1983. - .1. - 368.; .. - 416.
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F