2.
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:
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V. 1 A1, A2, A3, A4 , A4 A1A2A3.
V.
VI .
VII. M¢, M ( 1-15) ( 16-31).
MATHCAD
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- augment(A,B,C,) , - ;
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A,B,C,. .
7.
x, a0 + a1x + a2x2 +....,
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2. , .
3. Symbolic/Polynomial Coefficients (/
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3. , , , ( . 1 x).
4. Symbolics / Variable / Substitute ( /
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11.
substitute,
Symbolic (). substitute
, , , ( 11).
Given/Find
n m :
f1(x1,...,xm) = b1,, | fn (x1,...,xm) = bn | |
, | x1,...,xm , , |
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f(x) = b, | (2) |
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Given ;
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Find (x1,...,xm)
x1,...,xm.
, Boolean ( ). , , <Ctrl>+<=>. Find , , Find. .
13.
1.
: ={ 15,-20,-1}, = { 0,2,1}, , ={ 0,1,-1}, , = { 5,-3,2 }.
: , , , 31:
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, , :
= a · + b· + c ·
a, b, c , x { , , }. (1) () a, b, c. a, b, c,
Given/Find. :
A: =augment(p,q,r)=
, , . . , . A, : |A| = - 15.
A , , . Given/Find :
a:=0, b:=0, c:=0
Given
x=a · p + b · q + c · r
. , ( ). explicit, Symbolic ().
, , .
x = a × p + b × q + c × r explicit, a, b, c x = -6 × p +1× q + 3× r.
|
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2
: A(-2,4,6), B(0,2,-4), C(-6,8,-10).
: A,B,C 3´1:
A:= , B:= , B:= ,
, . , , . A,B,C , :
AB:=B - A , AC:=C - A ,
^ |
cos( , )=
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cos a = 1
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cos( , )=-1
3.
: A1(1,-1,2),A2 (2,1,2),A3(1,1,4),A4 (6,-3,8).
:
3´1:
A1:= , A2:= , A3:= , A4:= ,
, A1, , , , . :
A12:=A2-A1 , A13:=A3-A1 , A14:=A4-A1 ,
1/6 , . . , :
V:= |(A12 A13)·A14| 6
, 1/3 S ( A1A2A3) h, A4 A1A2A3, . ( D A1A2A3) ½ , , , : S:= |A12´A13| . : h:=ar w:top="1134" w:right="850" w:bottom="1134" w:left="1701" w:header="720" w:footer="720" w:gutter="0"/><w:cols w:space="720"/></w:sectPr></wx:sect></w:body></w:wordDocument>"> 3 .
4.
: P 1: x +2 y - 2 z - 7=0, P 2: x + y - 35=0.
:
, . n̅1 P 1, n̅2
̅
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cos(n̅1,n̅2 )=
, . P 1 P 2, :
P1:=x+2y-2z-7, P2:=x+y-35
coeffs, :
A:= P1 coeffs,x ,B:= P1 coeffs,y ,
C:= P1 coeffs,z ,
A, B, C, , . - A, B, C, , .. . . on error:
A1:=0 on error A1, B1 :=0 on error B1, C1:=0 on error C1.
, n̅1 :
n̅1:= ,
n̅2: |
AA:= P2 coeffs,x ,BB:= P2 coeffs,y ,
CC:= P2 coeffs,z ,
AA1:= 0 on error AA1, BB1:= 0 on error BB1, CC1:= 0 on error CC1.
, n̅2 :
^ |
^ |
(n̅1,n̅2 ) . :
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5.
: a 1:2 x - 3 y - 2 z +6=0, a 2: x - 3 y + z +3=0
:
a 1 a 2
a 1: A 1 x + B 1 y + C 1 z + D 1 = 0 | (1) |
a 2: AA 1 x + BB 1 y + CC 1 z + DD 1 = 0 | (2) |
, (1),(2) ( ) .
= =
- M0(x0; y0; z0), , ={q0; q1; q2} .
(, ) . , . s1 s2 :
s1:=2x-3y-2z+6, s2:=x-3y+z+3,
≪ coeffs, ≫ :
:= s1 coeffs,x , B:= s1 coeffs,y ,
C:= s1 coeffs,z .
A, B, C, , . - A, B, C, , .. . . on error:
A1:=0 on error A1, B1:=0 on error B1, C1:=0 on error C1
, :
n̅1:= ,
AA:= s2 coeffs,x , BB:= s2 coeffs,y ,
CC:= s2 coeffs,z ,
AA1:=0 on error AA1, BB1:=0 on error BB1, CC1:=0 on error CC1.
, :
n̅2:= ,
, : q:=n1 n2 .
M0, Given\Find:
Given
s1 = 0, s2 = 0.
M:=Find(x,y,z) .
X0:=M0 -3, y0:=M1 0, z0:=M2
. :
m:=q0 -9, l:=q1 -4, k:=q2 -3, g:=|x0| 3, h:=|y0| 0, v:=|z0| 0.
explicit:
a:=if , , if , , explificit, x0, m, g
b:=if , , if , , explificit, y0, l, h
c:=if , , if , , explificit, z0, k, v
M0 ( , ). , :
A=b explicit,a,b = , b=c explicit,b,c =
6 :) M (0,-3,-2), | x - 0.5 | = | y +1.5 | = | z -1.5 | . |
-1 |
) M (-2,0,3), a: 2 x - 2 y +10 z +1= 0
:
) 115
. 1.
M ', M
l, | K - | MM ', |
l. M | K, | |
M ' | . |
K, , K l, l.
M, K M1 3´1:
M:= , M:= , M1:= ,
M1 M¢. a,b, :
a: = x - 0.5, b = y + 1.5, c = z 1.5.
. 3´1 n:
n:=
M0Î l a,b,c (2), , . ≪ coeffs, ≫ :
A:= a coeffs,x , B:= b coeffs,y , C:= c coeffs,z ,
A, B, C, , . M0 :
M0:= ,
, M l, . , . K M, s:
MK:= , s:= MK×n
, :
s=0 z-y-1=0.
K, l, . Given\Find:
Given
a×n1=b×n0, b×n2=c×n1, s=0.
K:=Find(x,y,z)
n0,n1,n2- n. K MM1. (. 1, 2), M1 , Given\Find:
Given
=K
M1:=Find(x1,y1,z1))
) 1631.
M ', M a, K - M M ', a.
K MM ' a (. 13). , a
MM '. | M K, | ||||
M ' | (1). | ||||
M, K M1 | 3´1: | ||||
M:= , K:= , M1:= ,
M1 M ¢. MM '.
a. , . s:
s:=2 x - 2 y +10 z +1.
coeffs, :
A:= s coeffs,x , B:= s coeffs,y ,
C:= s coeffs,z ,
A, B, C, , . - A, B, C, , .. . . on error:
A1:= 0 on error A1, B1:= 0 on error B1, C1:= 0 on error C1.
, n̅ :
n:= ,
:
X0:=M0 -2, y0:=M1 0, z0:=M2
m:=n0 2, l:=n1 -2, k:=n2 10,
g:=|M0| 3, h:=|M1| 0, v:=|M2| 3.
explicit:
a:=if , , if , , explificit, x0, m, g
b:=if , , if , , explificit, y0, l, h
c:=if , , if , , explificit, z0, k, v
M 0 ( , ). ,
:
a=b explicit, a,b = , b=c explicit, c,b =
K, a, . Given\Find:
Given
s =0,l×(x - x0)=m×(y - y0),k×(y - y0)=l×(z - z0).
K:=Find(x,y,z)
K MM1. (. 1), M1 ,
Given\Find:
Given
=K
M1:=Find(x1,y1,z1)
. , ,
.1. x ={-2,4,7}, p ={0,1,2}, q ={1, 0,1}, r = {-1,2,4}.
1.2. x ={6,12,−1}, p ={1,3, 0}, q ={2,−1,1}, r ={0,−1,2}.
1.3. x ={1,−4,4}, p ={2,1,1}, q ={0,3, 2}, r ={1,−1,1}.
1.4. x ={-9,5,5}, p ={4,1,1}, q ={2, 0,−3}, r ={-1,2,1}.
1.5. x ={-5,-5,5}, p ={-2,0,1}, q ={1, 3,−1}, r ={0,4,1}.
1.6. x ={13,2,7}, p ={5,1, 0}, q ={2, −1, 3}, r ={1,0,−1}.
1.7. x ={-19,−1, 7}, p ={0,1, 1}, q ={-2,0, 1}, r ={3,1,0}.
1.8. x ={3,−3,4}, p ={1,0,2}, q ={0,1, 1}, r ={2,−1,4.}.
1.9. x ={3, 3,−1}, p ={3,1, 0}, q ={-1,2, 1}, r ={-1,0,2}.
1.10. x ={-1,7,−4}, p ={-1,2, 1}, q ={2,0, 3}, r ={1,1,−1}.
1.11. x ={6,5,−14}, p ={1,1, 4}, q ={0,−3, 2}, r ={2,1,−1}.
1.12. x ={6,−1,7}, p ={1,−2, 0}, q ={-1,1,3}, r ={1,0,4}.
1.13. x ={5,15, 0}, p ={1,0, 5}, q ={-1,3, 2}, r ={0,−1,1}.
1.14. x ={2,−1,11}, p ={1,1, 0}, q ={0, 1,−2}, r ={1,0, 3}.
1.15. x ={11,5,−3}, p ={1,0, 2}, q ={-1,0, 1}, r ={2,5,−3}
1.16. x ={8,0,5}, p ={2,0, 1}, q ={1,1, 0}, r ={4,1,2}.
1.17. x ={3,1,8}, p ={0,1, 3}, q ={1, 2,−1}, r ={2, 0,−1}.
1.18. x ={8,1,12}, p ={1, 2,−1}, q ={3,0,2}, r ={-1, 1,1}.
1.19. x ={-9,−8,−3}, p ={1,4,1}, q ={-3,2, 0}, r ={1,−1,2}.
1.20. x ={- 5,9,−13}, p ={0, 1,−2}, q ={3,−1,1}, r ={4,1,0}.
1.21. x ={-15,5,6}, p ={0,5,1}, q ={3, 2,−1}, r ={-1,1,0}.
1.22. x ={8,9,4}, p ={1,0, 1}, q ={0,−2, 1}, r ={1,3,0}.
1.23. x ={23,−14,−30}, p ={2,1,0}, q ={1,−1, 0}, r ={-3,2,5}.
1.24. x ={3,1,3}, p ={2,1, 0}, q ={1,0, 1}, r ={4,2,1}.
1.25. x ={-1,7,0}, p ={0,3, 1}, q ={1,−1, 2}, r ={2, −1,0}.
1.26. x ={11,−1,4}, p ={1,−1,2}, q ={3,2,0}, r ={-1,1,1}.
1.27. x ={-13,2,18}, p ={1,1, 4}, q ={-3,0, 2}, r ={1,2,−1}.
1.28. x ={0,−8,9}, p ={0,−2,1}, q ={3,1,−1}, r ={4,0,1}.
1.29. x ={8,−7,−13}, p ={0,1,5}, q ={3,−1, 2}, r ={-1,0,1}.
1.30. x ={2,7,5}, p ={1,0, 1}, q ={1,−2, 0}, r ={0,3,1}
. . .
V. 1 A1, A2, A3, A4 , A4 A1A2A3.
2.1.
2.2.
2.3.
2.4.
2.5.
2.6.
2.7.
2.8.
2.9.
2.10.
2.11.
2.12.
2.13.