4. pRQMAQ ZADANA KAK LINIQ PERESE^ENIQ DWUH PLOSKOSTEJ
8 |
A1x + B1y + C1z + D1 = 0 |
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A2x + B2y + C2z + D2 = 0: |
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PRQMOJ W PROSTRAN- |
|TU SISTEMU NAZYWA@T OB]IMI URAWNENIQMI |
STWE. pOSKOLXKU NAIBOLEE NAGLQDNYMI I UDOBNYMI DLQ ISPOLXZOWA- NIQ PRI RE[ENII ZADA^ QWLQ@TSQ KANONI^ESKIE I PARAMETRI^ESKIE URAWNENIQ PRQMOJ, TO WOZNIKAET NEOBHODIMOSTX PEREHODA OT OB]IH URAWNENIJ PRQMOJ K KANONI^ESKIM (PARAMETRI^ESKIM).
zADA^A 16. pRIWESTI OB]EE URAWNENIE PRQMOJ K KANONI^ESKOMU
WIDU |
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5x ; 2y + 3z ; 4 = 0 |
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zAPISATX PARAMETRI^ESKIE: |
URAWNENIQ PRQMOJ. |
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rE[ENIE. dLQ SOSTAWLENIQ URAWNENIJ PRQMOJ NEOBHODIMO ZNATX KOORDINATY TO^KI, PRINADLEVA]EJ PRQMOJ, I KOORDINATY NAPRAW- LQ@]EGO WEKTORA. rE[ENIE ZADA^I BUDET SOSTOQTX IZ DWUH \TAPOW.
A). nAHOVDENIE TO^KI M0(x0 y0 z0):
pRQMAQ ZADANA KAK LINIQ PERESE^ENIQ DWUH PLOSKOSTEJ I IMEET BES^ISLENNOE MNOVESTWO TO^EK, KOORDINATY KOTORYH UDOWLETWORQ@T SISTEME ( ): sISTEMA QWLQETSQ NEOPREDELENNOJ, TAK KAK W NEJ DWA URAWNENIQ, A NEIZWESTNYH TRI. oDNO IZ NEIZWESTNYH MOVNO ZADATX PROIZWOLXNO. pUSTX, NAPRIMER, z = 0:
tOGDA SISTEMA ( ) PRIMET WID |
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; 2y ; 4 = 0 |
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+ 2y ; 4 = 0: |
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x = 1: zNA^ENIE |
sKLADYWAQ URAWNENIQ, POLU^IM 8x ; 8 = 0 T.E. |
y POLU^IM, PODSTAWIW NAJDENNOE ZNA^ENIE x W ( ): |
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2y = 4 ; 3x 2y = 4 |
; 3 y = 1=2: |
iTAK, TO^KA |
M0 POLU^ENA: |
M0(1 1=2 0): |
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b). nAHOVDENIE NAPRAWLQ@]EGO WEKTORA.
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nAPRAWLQ@]IJ WEKTOR PRQMOJ ~s = |
fm n pg |
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PERPENDIKULQREN WEKTORAM NORMALEJ OBEIH PERE- |
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SEKA@]IHSQ PLOSKOSTEJ |
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~ |
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pO\TOMU |
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N1 = f5 ;2 3g I |
N2 = f3 2 ;5g: |
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W KA^ESTWE NAPRAWLQ@]EGO WEKTORA MOVNO WZQTX |
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WEKTOR, QWLQ@]IJSQ WEKTORNYM PROIZWEDENIEM |
rIS. 126. |
WEKTOROW NORMALEJ PLOSKOSTEJ. iTAK, |
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;2 3 = 4 |
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~s = [N1 |
N2] = 5 |
i + 34 |
j |
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2 |
;5 |
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zAPISYWAEM KANONI^ESKIE URAWNENIQ PRQMOJ
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x ; x0 |
= y ; y0 |
= z ; z0 |
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x ; 1 |
= y ; 1=2 |
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p |
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4 |
34 |
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oKON^ATELXNO |
x ; 1 |
= y ; 1=2 |
= z: |
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dLQ TOGO, ^TOBY ZAPISATX PARAMETRI^ESKIE URAWNENIQ, PRIRAWNQEM KAVDOE OTNO[ENIE K PARAMETRU t I POLU^IM SISTEMU PARAMETRI^ES- KIH URAWNENIJ
8 x = 2t + 1
>< y = 17t + 1=2
> z = 8t:
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4.2.2. wZAIMNOE RASPOLOVENIE PRQMYH W PROSTRANSTWE
pRQMYE W PROSTRANSTWE MOGUT BYTX PARALLELXNY, PERPENDIKULQR- NY, ILI OBRAZOWYWATX W OB]EM SLU^AE KAKOJ-TO UGOL. oTMETIM, ^TO WSE \TI ZADA^I SWODQTSQ K ZADA^E O WZAIMNOJ ORIENTACII NAPRAWLQ- @]IH WEKTOROW \TIH PRQMYH I PO\TOMU RE[A@TSQ SREDSTWAMI WEK- TORNOJ ALGEBRY. wO WSEH SLU^AQH PO DANNYM URAWNENIQM PRQMYH NEOBHODIMO OPREDELITX KOORDINATY IH NAPRAWLQ@]IH WEKTOROW.
1. nAHOVDENIE UGLA MEVDU PRQMYMI
pUSTX DANY DWE PRQMYE
l1 : |
x ; x1 = y ; y1 = z ; z1 |
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m1 |
n1 |
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l2 : |
x ; x2 = y ; y2 = z ; z2 : |
rIS. 127. |
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uGLOM MEVDU DWUMQ PRQMYMI NAZYWAETSQ UGOL MEVDU DWUMQ LU^AMI, WYHODQ]IMI IZ KAKOJ-LIBO TO^KI PROSTRANSTWA, PARALLELXNO DAN- NYM PRQMYM. iZ RISUNKA QSNO, ^TO UGOL MEVDU PRQMYMI ESTX UGOL MEVDU NAPRAWLQ@]IMI WEKTORAMI \TIH PRQMYH
~s1 = fm1 n1 p1g |
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~s2 = fm2 n2 p2g: |
zAPISYWAEM IZWESTNU@ FORMULU DLQ KOSINUSA UGLA MEVDU WEKTORAMI
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cos ' = cos(~s1 |
~s2) = |
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(~s1 ~s2) |
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j ~s1 jj ~s2 j |
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cos ' = |
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m1m2 + n1n2 + p1p2 |
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q |
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m12 + n12 + p12 |
m22 + n22 + p22 |
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zADA^A 17. |
nAJTI KOSINUS UGLA MEVDU PRQMYMI |
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8 x = 2t ; 1 |
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8 x = t + 3 |
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l1 : |
> y = ;2 |
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> y = 5t ; 1 |
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> z = ;4: |
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rE[ENIE. pRQMYE ZADANY PARAMETRI^ESKIMI URAWNENIQMI, KO\F-
FICIENTAMI PRI t QWLQ@TSQ KOORDINATY NAPRAWLQ@]IH WEKTOROW \TIH PRQMYH
~s1 |
= f2 0 3g |
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~s2 = f1 5 0g: |
nAHODIM KOSINUS UGLA MEVDU WEKTORAMI |
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cos ' = cos(~s1 ~s2) = |
(~s1 ~s2) |
= |
134 |
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j ~s1 jj ~s2 j |
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~s2:
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2 1 + 0 |
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+ 3 0 |
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0:56: |
p22 + 02 + 32 |
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p13 |
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p26 |
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p12 + 52 + 02 |
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zADA^A 18. |
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nAJTI KOSINUS UGLA NAKLONA PRQMOJ K OSI OY |
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8 x ; 2y + z ; 4 = 0 |
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< 3x + y + 2 = 0: |
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rE[ENIE. pRQMAQ ZADANA OB]IMI URAWNENIQMI, TO ESTX KAK LINIQ |
PERESE^ENIQ DWUH:PLOSKOSTEJ, WEKTORY NORMALI KOTORYH |
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N1 = f1 ;2 1g N2 = f3 1 0g: |
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nAPRAWLQ@]IJ WEKTOR NAHODIM KAK IH WEKTORNOE PROIZWEDENIE |
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;2 |
1 = f;1 3 7g: |
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~s = [N1 N2] = 1 |
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kOSINUS UGLA, KOTORYJ OBRAZUET |
POLU^ENNYJ |
WEKTOR S OSX@ OY |
OPREDELQETSQ KAK OTNO[ENIE WTOROJ KOORDINATY WEKTORA K EGO DLINE |
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cos |
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p |
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p3 |
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1 + 9 + 49 |
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2. pROWERKA USLOWIJ PARALLELXNOSTI I PERPENDIKULQRNOS- TI PRQMYH W PROSTRANSTWE
qSNO, ^TO NAPRAWLQ@]IE WEKTORA PARALLELXNYH PRQMYH KOLLI-
NEARNY ~s1 |
jj ~s2: |
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zAPISYWAEM USLOWIE KOLLINEARNOSTI \TIH WEKTOROW W KOORDINATNOJ |
FORME |
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m1 |
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= p1 |
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n2 |
p2 |
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rIS. 128. |
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nAPRAWLQ@]IE WEKTORA PERPENDIKULQRNYH PRQMYH TAKVE WZAIM- NO PERPENDIKULQRNY ~s1 ? zAPI[EM USLOWIE PERPENDIKULQR- NOSTI
(~s1 ~s2) = 0
m1m2 + n1n2 + p1p2 = 0:
rIS. 129.
zADA^A |
19. |
pRQMYE |
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x = 4t |
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+ 3 |
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z |
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l1 : |
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l2 : |
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< z = 5t + 2 |
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PARALLELXNY |
> |
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TAK KAK IH NAPRAWLQ@]IE WEKTORA |
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~s1 = f4 ;1 5g |
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KOLLINEARNY : |
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;8 |
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zADA^A |
20. |
pRQMYE |
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2 |
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l1 : |
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x = y + 1 |
= z ; |
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I l2 |
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x ; 4 |
= y ; 7 |
= |
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;1 |
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2 |
;5 |
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PERPENDIKULQRNY, TAK KAK WZAIMNO PERPENDIKULQRNY IH NAPRAWLQ@- |
]IE WEKTORA |
~s1 = f2 ;5 ;4g |
I ~s2 = f3 2 ;1g: |
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dEJSTWITELXNO, |
(~s1 ~s2) = 0 |
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2 3 + (;5) 2 + (;4) (;1) = 0 |
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0 = 0: |
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zADA^A |
21. |
pERESEKA@TSQ LI PRQMYE |
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l1 : |
x ; 1 |
= y ; 7 |
= z ; 5 |
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l2 : |
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x ; 6 |
= y + 1 |
= z |
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uKAVEM NAIBOLEE PROSTOJ SPOSOB RE[ENIQ ZADA^I. |
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iZ URAWNENIJ PRQMYH MY IMEEM KOORDINATY TO^EK M1(1 7 5) I
M2(6 ;1 0) I NAPRAWLQ@]IH WEKTOROW ~s1 = f2 1 4g I~s2 = f3 ;2 1g: |
eSLI WWESTI W RASSMOTRENIE WEKTOR ;;;!M1M2 = |
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f5 ;8 ;5g |
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TO PONQTNO, ^TO, ESLI PRQMYE PERESEKA@T- |
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SQ, TO ONI LEVAT W ODNOJ PLOSKOSTI, I TRI |
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WEKTORA ~s1 |
~s2 |
I ;;;!M1M2 BUDUT KOMPLA- |
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NARNY. zAPI[EM USLOWIE KOMPLANARNOSTI |
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(~s1 |
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;;;!M1M2) = 0: |
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rIS. 130. |
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2 |
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iLI W KOORDINATNOJ FORME |
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;2 |
1 = 0: |
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;8 |
;5 |
0 |
0: |
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wY^ISLQQ OPREDELITELX |
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POLU^IM TOVDESTWO |
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tAKIM OB |
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RAZOM, DANNYE PRQMYE W PROSTRANSTWE PERESEKA@TSQ.
3. nAHOVDENIE RASSTOQNIQ OT TO^KI DO PRQMOJ W PROSTRAN- STWE
zADA^A 22. nAJTI RASSTOQNIE OT TO^KI M(;3 1 2) DO PRQMOJ
x + 1 |
= y ; 1 |
= z + 6 |
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1 |
;4 |
2 |
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rE[ENIE. rASSTOQNIE OT TO^KI DO PRQMOJ ESTX DLINA PERPENDI- KULQRA, OPU]ENNOGO IZ DANNOJ TO^KI NA DANNU@ PRQMU@. wOSPOLXZU- EMSQ SLEDU@]EJ SHEMOJ RE[ENIQ.
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oBRAZUEM WEKTOR |
;;;!M0M SOEDINQ@]IJ DAN- |
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NU@ TO^KU M(;3 1 2) S TO^KOJ NA PRQMOJ |
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M0(;1 1 ;6), |
POLU^AEM WEKTOR |
;;;!M0M = |
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rIS. 131. |
f;2 0 ;8g: |
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;;;!M0M I ~s = f1 ;4 2g: iS- |
CTROIM PARALLELOGRAMM NA WEKTORAH |
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KOMOE RASSTOQNIE QWLQETSQ WYSOTOJ \TOGO PARALLELOGRAMMA I NAHO- DITSQ PO IZWESTNOJ FORMULE
S d = a
S - PLO]ADX PARALLELOGRAMMA NAHODITSQ S POMO]X@ WEKTORNOGO PROIZWEDENIQ WEKTOROW-STORON,
a - DLINA OSNOWANIQ PARALLELOGRAMMA, |
T.E. DLINA WEKTORA ~s: |
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S = j[~s ;;;!M0M]j = |
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= j32i + 4j ; |
8kj = |
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= q322 + 42 + (;8)2 = 4p69: |
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a = j~sj = q |
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= p |
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12 + (;4)2 + 22 |
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;;;!M0M] |
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4p |
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69 |
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iSKOMOE RASSTOQNIE |
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d = a |
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7:25: |
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zAME^ANIE. aNALOGI^NYM OBRAZOM RE[AETSQ ZADA^A O NAHOVDENII RASSTOQNIQ MEVDU DWUMQ PARALLELXNYMI PRQMYMI W PROSTRANSTWE.
iZ URAWNENIQ ODNOJ PRQMOJ BERUTSQ KOORDINATY TO^KI I DALEE NA- HODITSQ RASSTOQNIE OT \TOJ TO^KI DO WTOROJ PRQMOJ, KAK POKAZANO WY[E.
4.2.3. wZAIMNOE RASPOLOVENIE PRQMOJ I PLOSKOSTI
w ZADA^AH NA WZAIMNOE RASPOLOVENIE PRQMOJ S PLOSKOSTX@ W PRO- STRANSTWE PRIWLEKA@TSQ NAPRAWLQ@]IJ WEKTOR PRQMOJ I WEKTOR NOR- MALI PLOSKOSTI.
1. nAHOVDENIE UGLA MEVDU PRQMOJ I PLOSKOSTX@
pUSTX TREBUETSQ NAJTI UGOL MEVDU PRQMOJ
x ; x0 |
= y ; y0 |
= z ; z0 |
I PLOSKOSTX@ Ax + By + Cz + D = 0: |
m |
n |
p |
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uGLOM MEVDU PRQMOJ I PLOSKOSTX@ W PROSTRASTWE NAZYWAETSQ UGOL MEVDU \TOJ PRQMOJ I EE PROEKCIEJ NA \TU PLOSKOSTX.
iZ RISUNKA WIDNO, ^TO UGOL MEVDU PRQMOJ
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I PLOSKOSTX@ |
' DOPOLNQET UGOL |
MEV- |
DU NAPRAWLQ@]IM WEKTOROM PRQMOJ |
~s = |
f~ |
g |
o |
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m n p |
I WEKTOROM NORMALI PLOSKOSTI |
N = fA B Cg |
DO 90 . |
rIS. 132. |
sREDSTWAMI WEKTORNOJ ALGEBRY NAHODIM UGOL :
~
cos = (~N ~s) :
jNj j~sj
nO cos = cos(90o ; ') = sin ' PO\TOMU MOVNO ZAPISATX FORMULU DLQ NAHOVDENIQ SINUSA UGLA MEVDU PRQMOJ I PLOSKOSTX@ W WEKTORNOJ I KOORDINATNOJ FORME
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~ |
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jAm + Bn + Cpj |
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sin |
' = j(N ~s)j = |
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: |
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pA |
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pm |
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jNj j~sj |
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+ B |
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+ C |
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+ n + p |
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tAK KAK UGOL ' PRINIMAET ZNA^ENIQ OT 0o DO |
180o A SINUS TAKIH |
UGLOW WELI^INA POLOVITELXNAQ, TO W ^ISLITELE FORMULY POSTAWLEN ZNAK MODULQ.
zADA^A |
23. |
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nAJTI UGOL MEVDU PRQMOJ |
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x + 4 |
y |
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z + 1 |
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; 2y + 2z ; 5 = 0: |
2 |
= 3 |
= |
;6 |
I |
PLOSKOSTX@ x |
rE[ENIE. iZ USLOWIJ ZADA^I IMEEM : |
WEKTOR NORMALI PLOSKOSTI |
~ |
;2 2g I NAPRAWLQ@]IJ WEKTOR PRQMOJ ~s = f2 3 ;6g: zA- |
N = f1 |
PISYWAEM FORMULU DLQ NAHOVDENIQ SINUSA UGLA MEVDU PRQMOJ I |
PLOSKOSTX@ |
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sin ' = |
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j1 2 + (;2) 3 + 2 |
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(;6)j |
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= j |
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16j |
= 16: |
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q1 + (;2) + 2 q2 + 3 + (;6) |
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49 |
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2. uSLOWIQ PARALLELXNOSTI I PERPENDIKULQRNOSTI PRQMOJ I PLOSKOSTI W PROSTRANSTWE
iZ RISUNKA WIDNO, ^TO, ESLI PRQMAQ I PLOSKOSTX PARALLELXNY, TO |
IH HARAKTERISTI^ESKIE WEKTORA |
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N = fA B Cg I ~s = fm n pg WZA- |
IMNO PERPENDIKULQRNY |
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N ? ~s |
(N ~s) = 0 |
Am + Bn + Cp = 0:
rIS. 133.
eSLI PRQMAQ PERPENDIKULQRNA PLOSKOSTI, TO
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WEKTORA N = fA B Cg I ~s = fm n pg BU-
DUT KOLLINEARNY
3.
~ |
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A |
B |
C |
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N jj ~s |
) |
m |
= n |
= p |
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nAHOVDENIE TO^KI PERESE^ENIQ PRQMOJ S PLOSKOSTX@
zADA^A |
24. |
nAJTI TO^KU PERESE^ENIQ PRQMOJ |
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= y |
= z ; 2 |
S PLOSKOSTX@ 3x + 5y |
+ z |
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21 = 0: |
3 |
5 |
1 |
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rE[ENIE. dLQ TOGO, ^TOBY NAJTI TO^KU PERESE^ENIQ PRQMOJ S PLOS- |
KOSTX@ DOSTATO^NO NAJTI RE[ENIE SISTEMY, WKL@^A@]EJ W SEBQ URAW- |
NENIQ PRQMOJ I PLOSKOSTI |
8 x + 2 |
= y |
= z ; 2 |
> |
3 |
5 |
1 |
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< |
3x + 5y + z ; 21 = 0: |
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139 |
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dLQ RE[ENIQ SISTEMY KANONI^ESKIE URAWNENIQ PRQMOJ UDOBNEE PRED- |
STAWITX PARAMETRI^ESKI. tOGDA SISTEMA PRIMET WID |
8 yx == 53tt ; 2 |
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> z = t + 2 |
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< 3x + 5y + z ; 21 = 0: |
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> |
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pODSTAWLQEM ZNA^ENIQ x y z IZ PERWYH URAWNENIJ W URAWNENIE PLOS- |
KOSTI:I RE[AEM EGO OTNOSITELXNO PARAMETRA t. |
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3(3t ; 2) + 5(5t) + 1 |
(t + 2) |
; 21 = 0 OTKUDA 35t = 25 |
T.E. ZNA^ENIE PARAMETRA |
t, |
SOOTWETSTWU@]EE TO^KE PERESE^ENIQ |
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25 |
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PRQMOJ I PLOSKOSTI, RAWNO |
t = 35 |
= |
7: |
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pODSTAWIM \TO ZNA^ENIE W PERWYE TRI URAWNENIQ SISTEMY I POLU^IM |
KOORDINATY TO^KI PERESE^ENIQ |
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8 x = 3 7 |
2 = |
7 |
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25 |
19 |
< |
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25 |
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M 7 |
7 7 ! : |
> y = 5 |
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5 |
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19 |
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7 |
+ 2 = 7 |
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4.2.4. sME[ANNYE ZADA^I NA PRQMU@ I PLOSKOSTX W PRO- STRANSTWE
zADA^A 25. sOSTAWITX URAWNENIQ PRQMOJ, PROHODQ]EJ ^EREZ TO^-
KU M0(1 2 ;6) PERPENDIKULQRNO PLOSKOSTI 5x ; 4y + 3z ; 1 = 0: rE[ENIE. uRAWNENIQ PRQMOJ BEREM W WIDE
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= y ; y0 |
= z ; z0 : |
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n |
p |
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kOORDINATY TO^KI, ^EREZ KOTORU@ PROHODIT |
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PRQMAQ, IZWESTNY. w KA^ESTWE NAPRAWLQ@- |
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]EGO WEKTORA |
~s = fm n pg MOVET SLU- |
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VITX WEKTOR NORMALI DANNOJ PLOSKOSTI ~s = |
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= fA B Cg = f5 ;4 3g: |
rIS. 135. |
fm n pg = = N |
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tAKIM OBRAZOM, URAWNENIQ PRQMOJ PRIMUT WID |
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x ; 1 |
= y ; 2 = z + 6 |
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zADA^A 26. sOSTAWITX URAWNENIE PLOSKOSTI, PROHODQ]EJ ^EREZ TO^- KU M1(5 2 ;6) PERPENDIKULQRNO PRQMOJ
x + 3 = y ; 1 = z : ;1 4 ;3
rE[ENIE. uRAWNENIE PLOSKOSTI, PROHODQ]EJ ^EREZ TO^KU, PERPEN- DIKULQRNO WEKTORU IMEET WID
A (x ; x0) + B (y ; y0) + C (z ; z0) = 0:
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kOORDINATY TO^KI, ^EREZ KOTORU@ PROHODIT PLOSKOSTX, |
IZWESTNY. |
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kAK WIDNO IZ RISUNKA 135, WEKTOROM NORMALI PLOSKOSTI N =~ fA B Cg |
MOVET SLUVITX NAPRAWLQ@]IJ WEKTOR DANNOJ PRQMOJ |
N = ~s = |
f;1 4 ;3g: tAKIM OBRAZOM, URAWNENIE PLOSKOSTI |
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;1 (x ; 5) + 4 (y ; 2) ; 3 (z + 6) = 0 |
;x + 4y ; 3z ; 21 = 0 |
x ; 4y + 3z + 21 = 0:
zADA^A 27. sOSTAWITX URAWNENIE PLOSKOSTI, PROHODQ]EJ ^EREZ TO^KU M0(;3 2 5) I PRQMU@
8 x = ;2t ; 5
<> y = 4t + 3
> z = t + 2:
nAJTI OB_EM TREUGOLXNOJ:PIRAMIDY, KOTORU@ PLOSKOSTX OTSEKAET OT KOORDINATNOGO OKTANTA.
rE[ENIE. iZ URAWNENIJ PRQMOJ MY MOVEM IZWLE^X SLEDU@]U@
INFORMACI@: |
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TO^KA NA PRQMOJ M1(;5 3 2) |
PRQ- |
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NAPRAWLQ@]IJ |
WEKTOR |
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MOJ ~s = |
f;2 4 1g: |
eSLI WWESTI W RASSMOT- |
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RENIE WEKTOR ;;;!M0M1 = f;2 1 ;3g |
TO WMES- |
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TE S NAPRAWLQ@]IM WEKTOROM PRQMOJ ~s = |
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f;2 4 1g |
MY BUDEM IMETX DWA WEKTORA, LE- |
rIS. 136. |
VA]IE W ISKOMOJ PLOSKOSTI. |
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