VECTO TRONG KHONG GIAN.
QUAN HE VUDNG GOC TRONG
KH6NG GIAN
• ; ' • • . •
I I I-I 1111
II
C* Vectd trong khong gian
*t* Hai dudng thing vuong goc
*t* Dudng thang vuong gdc vdi mdt phang
*** Hai mat phang vuong goc
*t* Khoang each
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Trong chuong nay chung ta se nghien cufu ve vecto
trong I ki, goi B vd C Id hai diim
'1,1
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,
,
if! sao cho AB = U, AC = v.
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Khi dd ta ggi gdc BAC
(0° < BAC < 180°) Id gdc
giita hai vecta U vd v
trong khdng gian, ki hiiu la
{u,v) (h.3.11).
Hinh 3.11
^ 1 Cho tfl di6n diu ABCD co H Id trung dilm cua canh AB. Hay tfnh goc giflacac cap
vecto sau ddy:
a) AB vd BC ;
b) C ^ va AC.
2. Tich vd hudng cua hai vecta trong khdng gian
\ Djnh nghla
• khdng.
. Trong khdng gian cho hai vecta u vd v diu khdc vecta
•
"' Tich vd hudng cua hai vecta U vd v Id mot so, ki hiiu Id
,', it. V, duac xdc dinh bdi cdng thitc :
M.v =|M|.|i^|.eos(i<,v)
Trudng hgp M = 0 hoae i' = 0 ta quy udc M.V = 0.
Vi du 1. Cho tfl dien OABC cd cdc canh OA, OB, OC ddi mdt vudng gdc vd
OA = OB = OC = 1. Ggi M Id trung dilm eua canh AB. Tinh gdc gifla hai
vecto OM vd BC.
93
giai
Ta cd cos {OM, BC) = ^E:^^
\OM\.\BC\
OM.BC
(h.3.12).
Mat khdc OM.BC = -{oA + OB\.{OC - OB)
= - {OA.OC - OA.OB + OB.OC - OB )
Vi OA, OB, OC ddi mdt vudng gde va OB = 1 ndn
OAIOC
Do dd cos {OM, ^)
= ---
= dAm
= 08.00 = Ovk OB =1.
Vdy {OM,BC) = 120°.
A 2 Cho hinh lap phuong ABCD.A'B'C'D'.
a) Hay phan tfch cdc vecto AC' vd BD theo ba vecto AB, AD, AA'.
b) Tfnh cos (AC', BD) vd tfl do suy ra AC' vd BD vudng goc vdi nhau.
II. VECTO CHI PHl/ONG CUA D U 6 N G THANG
/ . Dinh nghia
j | Vecta a khdc vecta - khdng duac
ggi Id vecta chi phuang ciia dudng
thdng d niu gid cua vecta a song d_
song hodc triing vdi dudng thdng d
(h.3.13).
_,
Hinh 3.13
2. Nhgn xet
a) Ne'u d la vecto chi phuang cua dudng thing d thi vecto ka vdik ^ 0 cung
la vecto chi phuong eua d.
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b) Mdt dudng thing d trong khdng gian hodn todn duge xde dinh nlu bie't mdt
dilm A thude d vk mdt vecto chi phuong a cua nd.
c) Hai dudng thing song song vdi nhau khi vd chi khi chflng la hai dudng
thing phdn biet vd cd hai vecta chi phuong cung phuong.
HI. GOC GI0A HAI D U 6 N G THANG TRONG KHONG GIAN
Trong khdng gian cho hai dudng thing a, b bd't ki. Tfl mdt dilm O nao dd
ta ve hai dudng thing a' va fe' ldn Iugt song song vdi a vdfe.Ta nhdn thd'y
ring khi dilm O thay ddi thi gdc gifla a' vkb' khdng thay ddi. Do dd ta cd
dinh nghia :
/. Dinh nghla
ly Gdc giita hai dudng thdng avdb trong khdng gian Id gdc giita
|i hai dudng thdng a' vd b' cung di qua mgt diim vd ldn luat
|| song song vdi avdb (h.3.14).
O
Hinh 3.14
2. Nhdn xet
a) Dl xdc dinh gde gifla hai dudng thing a vdfeta ed thi ldy dilm O thude
mdt trong hai dudng thing dd rdi ve mdt dudng thing qua O vd song song vdi
dudng thing edn lai.
b) Nlu M Id vecto ehi phuong eua dudng thing a va v Id vecto ehi phuang
cua dudng thingfevd {U,v) = or thi gdc gifla hai dudng thing a vdfebing a
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