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Ï°Ìâ7-1
1. Éèu=a?b+2c, v=?a+3b?c. ÊÔÓÃa¡¢b¡¢c±íʾ2u?3v .
½â 2u?3v =2(a?b+2c)?3(?a+3b?c)=2a?2b+4c+3a?9b+3c=5a?11b+7c .
2. Èç¹ûƽÃæÉÏÒ»¸öËıßÐεĶԽÇÏß»¥Ïàƽ·Ö, ÊÔÓÃÏòÁ¿Ö¤Ã÷ÕâÊÇƽÐÐËıßÐÎ. Ö¤Ã÷ ; , ¶ø, ,
ËùÒÔ.
Õâ˵Ã÷ËıßÐÎABCDµÄ¶Ô±ßAB=CDÇÒAB//CD, ´Ó¶øËıßÐÎABCDÊÇƽÐÐËıßÐÎ.
3. °Ñ¦¤ABCµÄBC±ßÎåµÈ·Ö, Éè·ÖµãÒÀ´ÎΪD¡¢D¡¢D¡¢D, ÔٰѸ÷·ÖµãÓëµãAÁ¬½Ó. ÊÔÒÔ¡¢
1
2
3
4
±íʾÏòÁ¿¡¢¡¢A3¡¢A4. ½â
, , ,
.
4. ÒÑÖªÁ½µãM(0, 1, 2)ºÍM(1, ?1, 0). ÊÔÓÃ×ø±ê±íʾʽ±íʾÏòÁ¿¼°.
1
2
½â , .
5. ÇóƽÐÐÓÚÏòÁ¿a=(6, 7, ?6)µÄµ¥Î»ÏòÁ¿. ½â
,
ƽÐÐÓÚÏòÁ¿a=(6, 7, ?6)µÄµ¥Î»ÏòÁ¿Îª »ò . 6. ÔÚ¿Õ¼äÖ±½Ç×ø±êϵÖÐ, Ö¸³öÏÂÁи÷µãÔÚÄĸöØÔÏÞ£¿ A(1, ?2, 3); B(2, 3, ?4); C(2, ?3, ?4); D(?2, ?3, 1).
½â AÔÚµÚËÄØÔÏÞ, BÔÚµÚÎåØÔÏÞ, CÔÚµÚ°ËØÔÏÞ, DÔÚµÚÈýØÔÏÞ.
7. ÔÚ×ø±êÃæÉϺÍ×ø±êÖáÉϵĵãµÄ×ø±ê¸÷ÓÐʲôÌØÕ÷£¿Ö¸³öÏÂÁи÷µãµÄλÖÃ:
A(3, 4, 0); B(0, 4, 3); C(3, 0, 0); D(0, ?1, 0).
½â ÔÚxOyÃæÉÏ, µÄµãµÄ×ø±êΪ(x, y, 0); ÔÚyOzÃæÉÏ, µÄµãµÄ×ø±êΪ(0, y, z); ÔÚzOxÃæÉÏ, µÄµãµÄ×ø±êΪ(x, 0, z).
ÔÚxÖáÉÏ, µÄµãµÄ×ø±êΪ(x, 0, 0); ÔÚyÖáÉÏ, µÄµãµÄ×ø±êΪ(0, y, 0), ÔÚzÖáÉÏ, µÄµãµÄ×ø±êΪ(0, 0, z).
AÔÚxOyÃæÉÏ, BÔÚyOzÃæÉÏ, CÔÚxÖáÉÏ, DÔÚyÖáÉÏ.
8. Çóµã(a, b, c)¹ØÓÚ(1)¸÷×ø±êÃæ; (2)¸÷×ø±êÖá; (3)×ø±êÔµãµÄ¶Ô³ÆµãµÄ×ø±ê.
½â (1)µã(a, b, c)¹ØÓÚxOyÃæµÄ¶Ô³ÆµãΪ(a, b, ?c); µã(a, b, c)¹ØÓÚyOzÃæµÄ¶Ô³ÆµãΪ(?a, b, c); µã(a, b, c)¹ØÓÚzOxÃæµÄ¶Ô³ÆµãΪ(a, ?b, c).
(2)µã(a, b, c)¹ØÓÚxÖáµÄ¶Ô³ÆµãΪ(a, ?b, ?c); µã(a, b, c)¹ØÓÚyÖáµÄ¶Ô³ÆµãΪ(?a, b, ?c); µã(a, b, c)¹ØÓÚzÖáµÄ¶Ô³ÆµãΪ(?a, ?b, c).
(3)µã(a, b, c)¹ØÓÚ×ø±êÔµãµÄ¶Ô³ÆµãΪ(?a, ?b, ?c).
9. ×ÔµãP(x, y, z)·Ö±ð×÷¸÷×ø±êÃæºÍ¸÷×ø±êÖáµÄ´¹Ïß, д³ö¸÷´¹×ãµÄ×ø±ê.
0
0
0
0
½â ÔÚxOyÃæ¡¢yOzÃæºÍzOxÃæÉÏ, ´¹×ãµÄ×ø±ê·Ö±ðΪ(x, y, 0)¡¢(0, y, z)ºÍ(x, 0, z).
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0
0
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0
ÔÚxÖá¡¢yÖáºÍzÖáÉÏ, ´¹×ãµÄ×ø±ê·Ö±ðΪ(x, 0, 0), (0, y, 0)ºÍ(0, 0, z).
0
0
10. ¹ýµãP(x, y, z)·Ö±ð×÷ƽÐÐÓÚzÖáµÄÖ±ÏߺÍƽÐÐÓÚxOyÃæµÄƽÃæ, ÎÊÔÚËüÃÇÉÏÃæµÄµã
0
0
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0
µÄ×ø±ê¸÷ÓÐʲôÌص㣿
½â ÔÚËù×÷µÄƽÐÐÓÚzÖáµÄÖ±ÏßÉÏ, µãµÄ×ø±êΪ(x, y, z); ÔÚËù×÷µÄƽÐÐÓÚxOyÃæµÄƽÃæÉÏ,
0
0
µãµÄ×ø±êΪ(x, y, z).
0
11. Ò»±ß³¤ÎªaµÄÁ¢·½Ìå·ÅÖÃÔÚxOyÃæÉÏ, Æäµ×ÃæµÄÖÐÐÄÔÚ×ø±êÔµã, µ×ÃæµÄ¶¥µãÔÚxÖáºÍyÖáÉÏ, ÇóËü¸÷¶¥µãµÄ×ø±ê. ½â ÒòΪµ×ÃæµÄ¶Ô½ÇÏߵij¤Îª
,
,
, ËùÒÔÁ¢·½Ìå¸÷¶¥µãµÄ×ø±ê·Ö±ðΪ
,
,
, , , . 12. ÇóµãM(4, ?3, 5)µ½¸÷×ø±êÖáµÄ¾àÀë.
½â µãMµ½xÖáµÄ¾àÀë¾ÍÊǵã(4, ?3, 5)Óëµã(4, 0, 0)Ö®¼äµÄ¾àÀë, ¼´ .
µãMµ½yÖáµÄ¾àÀë¾ÍÊǵã(4, ?3, 5)Óëµã(0, ?3, 0)Ö®¼äµÄ¾àÀë, ¼´ .
µãMµ½zÖáµÄ¾àÀë¾ÍÊǵã(4, ?3, 5)Óëµã(0, 0, 5)Ö®¼äµÄ¾àÀë, ¼´
.
13. ÔÚyOzÃæÉÏ, ÇóÓëÈýµãA(3, 1, 2)¡¢B(4, ?2, ?2)ºÍC(0, 5, 1)µÈ¾àÀëµÄµã. ½â ÉèËùÇóµÄµãΪP(0, y, z)ÓëA¡¢B¡¢CµÈ¾àÀë, Ôò
, , .
ÓÉÌâÒâ, ÓÐ ,
¼´ ½âÖ®µÃy=1, z=?2, ¹ÊËùÇóµãΪ(0, 1, ?2).
14. ÊÔÖ¤Ã÷ÒÔÈýµãA(4, 1, 9)¡¢B(10, ?1, 6)¡¢C(2, 4, 3)Ϊ¶¥µãµÄÈý½ÇÐÎÊǵÈÑüÈý½ÇÖ±½ÇÈý½ÇÐÎ.
½â ÒòΪ
ËùÒÔ, .
Òò´Ë¦¤ABCÊǵÈÑüÖ±½ÇÈý½ÇÐÎ. 15. ÉèÒÑÖªÁ½µã ½â
,
,
;
;
2
, ,
,
ºÍM(3, 0, 2). ¼ÆËãÏòÁ¿µÄÄ£¡¢·½ÏòÓàÏҺͷ½Ïò½Ç.
;
, , .
16. ÉèÏòÁ¿µÄ·½ÏòÓàÏÒ·Ö±ðÂú×ã(1)cos¦Á=0; (2)cos¦Â=1; (3)cos¦Á=cos¦Â=0, ÎÊÕâЩÏòÁ¿Óë×ø±êÖá»ò×ø±êÃæµÄ¹ØϵÈçºÎ£¿
½â (1)µ±cos¦Á=0ʱ, ÏòÁ¿´¹Ö±ÓÚxÖá, »òÕß˵ÊÇƽÐÐÓÚyOzÃæ. (2)µ±cos¦Â=1ʱ, ÏòÁ¿µÄ·½ÏòÓëyÖáµÄÕýÏòÒ»ÖÂ, ´¹Ö±ÓÚzOxÃæ.
(3)µ±cos¦Á=cos¦Â=0ʱ, ÏòÁ¿´¹Ö±ÓÚxÖáºÍyÖá, ƽÐÐÓÚzÖá, ´¹Ö±ÓÚxOyÃæ. 17. ÉèÏòÁ¿rµÄÄ£ÊÇ4, ËüÓëÖáuµÄ¼Ð½ÇÊÇ60¡ã, ÇórÔÚÖáuÉϵÄͶӰ. ½â .
18. Ò»ÏòÁ¿µÄÖÕµãÔÚµãB(2, ?1, 7), ËüÔÚxÖá¡¢yÖáºÍzÖáÉϵÄͶӰÒÀ´ÎΪ4, ?4, 7. ÇóÕâÏòÁ¿µÄÆðµãAµÄ×ø±ê.
½â ÉèµãAµÄ×ø±êΪ(x, y, z). ÓÉÒÑÖªµÃ
,
½âµÃx=?2, y=3, z=0. µãAµÄ×ø±êΪA(?2, 3, 0).
19. Éèm=3i+5j+8k, n=2i?4j?7kºÍp=5i+j?4k. ÇóÏòÁ¿a=4m+3n?pÔÚxÖáÉϵÄͶӰ¼°ÔÚyÖáÉϵķÖÏòÁ¿.
½â ÒòΪa=4m+3n?p=4(3i+5j+8k)+3(2i?4j?7k)?(5i+j?4k )=13i+7j+15k, ËùÒÔa=4m+3n?pÔÚxÖáÉϵÄͶӰΪ13, ÔÚyÖáÉϵķÖÏòÁ¿7j .
Ï°Ìâ7?2
1. Éèa=3i?j?2k, b=i+2j?k, Çó(1)a?b¼°a¡Áb; (2)(?2a)?3b¼°a¡Á2b; (3)a¡¢b¼Ð½ÇµÄÓàÏÒ.
½â (1)a?b=3¡Á1+(?1)¡Á2+(?2)¡Á(?1)=3,
. (2)(?2a)?3b =?6a?b = ?6¡Á3=?18, a¡Á2b=2(a¡Áb)=2(5i+j+7k)=10i+2j+14k .
(3) .
2. Éèa¡¢b¡¢cΪµ¥Î»ÏòÁ¿, ÇÒÂú×ãa+b+c=0, Çóa?b+b?c+c?a . ½â ÒòΪa+b+c=0, ËùÒÔ(a+b+c)?(a+b+c)=0, ¼´ a?a+b?b+c?c+2a?b+2a?c+2c?a=0, ÓÚÊÇ
1
2
3
.
3. ÒÑÖªM(1, ?1, 2)¡¢M(3, 3, 1)ºÍM(3, 1, 3). ÇóÓ롢ͬʱ´¹Ö±µÄµ¥Î»ÏòÁ¿. ½â , .
,
,
ΪËùÇóÏòÁ¿.
4. ÉèÖÊÁ¿Îª100kgµÄÎïÌå´ÓµãM(3, 1, 8)ÑØÖ±Ï߳ƶ¯µ½µãM(1, 4, 2), ¼ÆËãÖØÁ¦Ëù×÷µÄ¹¦(³¤
1
2
¶Èµ¥Î»Îªm, ÖØÁ¦·½ÏòΪzÖḺ·½Ïò).
½âF=(0, 0, ?100¡Á9. 8)=(0, 0, ?980), .
W=F?S=(0, 0, ?980)?(?2, 3, ?6)=5880(½¹¶ú).
5. ÔڸܸËÉÏÖ§µãOµÄÒ»²àÓëµãOµÄ¾àÀëΪxµÄµãP´¦, ÓÐÒ»Óë³É½Ç¦È
1
1
1
µÄÁ¦F×÷ÓÃ×Å;
1
ÔÚOµÄÁíÒ»²àÓëµãOµÄ¾àÀëΪxµÄµãP´¦, ÓÐÒ»Óë³É½Ç¦È
2
2
1
µÄÁ¦F×÷ÓÃ×Å. Îʦȡ¢¦È¡¢x¡¢
1
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1
x¡¢|F|¡¢|F|·ûºÏÔõÑùµÄÌõ¼þ²ÅÄÜʹ¸Ü¸Ë±£³Öƽºâ£¿
2
1
2
½â ÒòΪÓй̶¨×ªÖáµÄÎïÌåµÄƽºâÌõ¼þÊÇÁ¦¾ØµÄ´úÊýºÍΪÁã, ÔÙ×¢Òâµ½¶ÔÁ¦¾ØÕý¸ºµÄ¹æ¶¨¿ÉµÃ, ʹ¸Ü¸Ë±£³ÖƽºâµÄÌõ¼þΪ x|F|?sin¦È?x|F|?sin¦È=0,
11
11
11
22
22
22
¼´ x|F|?sin¦È=x|F|?sin¦È.
6. ÇóÏòÁ¿a=(4, ?3, 4)ÔÚÏòÁ¿b=(2, 2, 1)ÉϵÄͶӰ.
½â . 7. Éèa=(3, 5, ?2), b=(2, 1, 4), ÎʦËÓë¦ÌÓÐÔõÑùµÄ¹Øϵ, ÄÜʹµÃ¦Ëa+¦ÌbÓëzÖá´¹Ö±? ½â ¦Ëa+¦Ìb=(3¦Ë+2¦Ì, 5¦Ë+¦Ì, ?2¦Ë+4¦Ì), ¦Ëa+¦ÌbÓëzÖá´¹?¦Ëa+¦Ìb ¡Ík
?(3¦Ë+2¦Ì, 5¦Ë+¦Ì, ?2¦Ë+4¦Ì)?(0, 0, 1)=0, ¼´?2¦Ë+4¦Ì=0, ËùÒÔ¦Ë=2¦Ì . µ±¦Ë=2¦Ì ʱ, ¦Ëa+¦ÌbÓëzÖá´¹Ö±. 8. ÊÔÓÃÏòÁ¿Ö¤Ã÷Ö±¾¶Ëù¶ÔµÄÔ²ÖܽÇÊÇÖ±½Ç. Ö¤Ã÷ ÉèABÊÇÔ²OµÄÖ±¾¶, CµãÔÚÔ²ÖÜÉÏ, Ôò, . ÒòΪ,
ËùÒÔ, ¡ÏC=90¡ã.
9. ÉèÒÑÖªÏòÁ¿a=2i?3j+k, b=i?j+3kºÍc=i?2j, ¼ÆËã: (1)(a?b)c?(a?c)b; (2)(a+b)¡Á(b+c); (3)(a¡Áb)?c . ½â (1)a?b=2¡Á1+(?3)¡Á(?1)+1¡Á3=8, a?c=2¡Á1+(?3)¡Á(?2)=8,
(a?b)c?(a?c)b=8c?8b=8(c?b)=8[(i?2j)?(i?j+3k)]=?8j?24k . (2)a+b=3i?4j+4k, b+c=2i?3j+3k,
.
(3) , (a¡Áb)?c=?8¡Á1+(?5)¡Á(?2)+1¡Á0=2.
10. ÒÑÖª, , Çó¦¤OABµÄÃæ»ý.
½â ¸ù¾ÝÏòÁ¿»ýµÄ¼¸ºÎÒâÒå, ±íʾÒÔºÍΪÁڱߵÄƽÐÐËıßÐεÄÃæ»ý, ÓÚÊǦ¤OABµÄÃæ»ýΪ