chapter 02 - 向量
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By: Anthony Bedford and Wallace Fowler
Engineering
Mechanics:
STATICS
Fifth Edition
in SI Units
Chapter 2:Chapter 2: VectorsVectors
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Chapter 2 Engineering Mechanics: STATICS
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學習目標
如果一個物體受到不同大小與不同方向的外力作用時,如何能決定作用於個物體
!"力的大小#方向$%
&'()*+向,的-.,如何/0,123向,,與43向,&567用8的9:;
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'<大=
• >,與向,
• ?@0,
• A@0,
• >,B
• 向,B
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• –>, CDEFGHI的物J,
– 9如K時L,M,
• –向, NO大小PQRFGST方向
– 9如KULV一個WXY於Z一W的[\,作用力
– E]體^_23K U, V, W, …
– 向, U 的大小 = |U|
>,#向,
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– 向,的`a23Kbc
• bc方向 = 向,方向
• bcde 向,大小
• 9如K
– r AB = W B XY於W A 的[\向,
– r AB 的方向 = fW A 到W B 的方向
>,#向,
∝
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>,#向,
– |r AB| = gW!L的hi
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Chapter 2 Engineering Mechanics: STATICS
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>,#向,
j向,kl "
mn一個物體 受到[o # fUL一[\op到qr[\$
– [o向,K U
– U 的方向 = [o的方向
– |U| = sop的hi
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Chapter 2 Engineering Mechanics: STATICS
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– t?個[o V
– suvw[\x一y,z{| U } V ,~X•€;
– U T V ‚ƒ於„一[o
W: U + V = W
>,#向,
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Chapter 2 Engineering Mechanics: STATICS
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>,#向,
j向,kl的定… "
– 向,f†‡ U 到ˆ V
jA‰a定Š
mk‹與向,LŒ†X的€zŽ;
j‘’a定Š
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Chapter 2 Engineering Mechanics: STATICS
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>,#向,
m向,kl“NO”–的K
U + V = V + U #2'&$
m向,“klNO—"–的 "
(U + V) + W = U + (V + W) #2'2$
m如果g個~/8的向,的#‚於˜,™š›一個œ的ž’a;
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Chapter 2 Engineering Mechanics: STATICS
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>,#向,
– 9如 "
jf A 到 C 的向, r AC ‚於
r AB 與 r BC 的向,#;
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Chapter 2 Engineering Mechanics: STATICS
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• >,與向,的ŸBK
– >, # FG $ a 與向, U 的ŸB = 向, aU
– 大小 = |a||U| ,qV |a| >, a 的¡Y¢
– aU 的方向同於 U ,n a £
– aU 的方向•於 U ,n a R
>,#向,
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Chapter 2 Engineering Mechanics: STATICS
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>,#向,
m向, U ¤/>, a ¥定…如¦K
UU
=
aa
1
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Chapter 2 Engineering Mechanics: STATICS
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>,#向,
mŸBY>,的ŸlN—"–K
a(bU) = (ab)U #2'3$
mŸBY>,klN0§–K
(a + b)U = aU + bU #2'4$
mŸBY向,klN0§–Ka(U + V) = aU + aV #2'5$
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Chapter 2 Engineering Mechanics: STATICS
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>,#向,
j向,¨lK
U – V = U + (−1)V #2'$
j„[向,K
mde‚於 1
m©定一個方向
m如果„[向, e /T向, U 的方向X同K U = |U|e
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ª9 2.1
向,的-.
«3!向,的de“ |U| = 8 與 |V| = 3 ; 向, V “¬的;®用`¯的方°決定向, U + 2V 的de;
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方±
²³9´µg個向,}-用向,kl的A‰a定Š,()能,µ向, U + 2V 的de;
ª9 2.1( ¶ )
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¯·
²³9´µ U # 2V ,¸¹)Œ†X;
ª9 2.1( ¶ )
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¯·
|U + 2V| º,的¢“ 13.0 ;
ª9 2.1( ¶ )
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»習¼
«3的向,的de“ |U| = 8 與 |V| = 3 ;向, V “¬ 的; ®/`¯的方°,µ向, U − 2V 的de;
·½K |U − 2V| = 5.7
ª9 2.1( ¶ )
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ª9 2.2
向,Xk
( ¾¿Àu )
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?@0,
jn向,/XÁ¬的0,123時,向,ÂÃÄÅÆJ
ÞK
– ÇÈ向, U K
mÉ\‰Ê標,Ë向, U 與 x-y Ì
– U = 0Í於 x Î# Ï Î的X¬0, U x # U y
的#K U = U x + U y
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mÐÑ£ x Î方向的„[向, i #£ y Î方向的„[向
, j"
U = U xi + U y j #2'!$
qV U x 與 U y “ U 的的>,0,;
– U 的de用ÒÓÔ定JE¹的0,.µ "
#2'8$
?@0,
22
y xU U +=U
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?@0,
j/0,作向,-.K
– g個向, U 與 V 的# "
U + V = (U xi + U y j) + (V xi + V y j)
= (U x + V x)i + (U y + V y) j #2'%$
– 如` "
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?@0,
j/0,作向,-. "
– G^ a 與向, U 的ŸB“ "
aU = a(U xi + U y j) = aU xi + aU y j
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?@0,
j/向,0,23[\向, "
mÇÈÊ標 # x A, y A$ !W A /TÊ標 # x B, y B$ !W
B ;
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?@0,
j/向,0,23[\向, "
– Õ r AB Ö2 B XY於 A 的[\向,K
r AB = ( x B − x A)i + ( y B − y A) j #2'&0$
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ª9 2.3
×.0,
ØÙfW A 到W B ÚkÛ 900-N 的力於ÜÝÞßà的áâã/向, ( Ö2;
®ä ( /«3!Ê
標åæ¦的0,123;
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ª9 2.3( ¶ )
方±
()ä用gç方°×.向, F 的0,;t一個方l,() äË用A‰學×. F 與 Y Î!L的è‰1.µ0,;t?
個方l,()äË用éê的ØÙ AB 的ëìãË用XíA
‰a1×. F 的0,;
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¯·
t一個方l
×. F 與 y Ê標Î的è‰K
°=
= 62680
40arcta .α
ª9 2.3( ¶ )
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¯·
î用A‰ï何學×.µ F ãð¹的0,23°K
( )
( ) N805402
N6.26c!"9006.26"#900c!""#
!!
ji
ji
jFiFF
−=−=
−= α α
ª9 2.3( ¶ )
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¯·
t?個方l
®Ë用«ñ定的òó,×.f A
到 B 的hiK
( ) ( ) $489$80$4022
.=+
ª9 2.3( ¶ )
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¯·
î用XíA‰a×. F 的0,K
$89.4
$80 a%
$89.4
$40
== FF y x
F F
( ) ( )
( ) N805402
N9004.89
40 N900
4.89
40
ji
jiF
−=
−=
ª9 2.3( ¶ )
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»習¼
ØÙfW A 到W B ÚkÛ 900-N 的力於ÜÝÞßà的àáã/向, F 1Ö2;ô定õöpW B 的[\ËÃ F 的 y
0,的大小“ x 0,÷的 3 ø;®ä F /¹的0,123
;W B ùú& x Î8iûWžü的ý方%
·½K F = 285i - 854 j (N) ;W B ùú& x Î8iûW 26.7
$ !Æ;
ª9 2.3( ¶ )
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ª9 2.4
f‰e×.0,
( ¾¿Àu )
ª9 2.5×.þê向,的de
( ¾¿Àu )
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A@0,
& *+´µA@V的物體K
(a) ÝÙ¬於ÿ方體一Ì時«見到的ÿ方體;
(') ÿ方體的ë側Ý;
(c) 一個‰Ê標åæ與ÿ方體的’疊";
(%) Ê標åæ的A@23;
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j一個右手Ê標åæK
m/0,
U x) U y * U z #
0Í於 x) y 與 z Î $ 23 向, U K
U = U x + U y + U z #2'&&$
A@0,
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A@0,
mË用©向 x , y # 方向的„[向, i 、 j # k ,
()可/ä U î用>,0,23如¦
– U = U xi + U y j + U z k #2'&2$
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A@0,
j/向,0,23向,的deK
jÇÈ一個向, U /T¹的0,K
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A@0,
j/向,0,23向,的deK
mE U y , U z /Tg÷的‹# U y + U z «構›的‰A
‰aK
|U y + U z |2 = |U y|2 + |U z |2 #2'&3$
m向, U ‚於向, U x 與 U y + U z !#; 3 個向,
a›一個‰A‰a K
|U|2 = |U x|2 + |U y + U z |2
@0,
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A@0,
j/向,0,23向,的deK
mä° (2.13) Ö入K
|U|2 = |U x|2 + |U y| + |U z |2 = U x2 + U y2 + U z 2
m因此,向, U 的大小K
#2'&4$222
z y x U U U ++=U
@0,
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A@0,
j方向餘ÔK
– HI一個向,方向的方l!一,“標3µ向,與Ê標ΣÎ向!L的è‰ θ x)、θ y 與 θ z ;
U x = |U| c!" θ x,
U y = |U| c!" θ y,
U z = |U| c!" θ z #2'&5$
A@0,
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A@0,
& 方向餘ÔK
A@0,
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A@0,
& 方向餘ÔK
– 方向餘Ô K c!" θ x) c!" θ y * c!" θ z
– 方向餘Ô滿足¦IŽ係°K
c!"2 θ x + c!"2 θ y + c!"2 θ z = 1 #2'&$
– ô定 e “一個與 U 同方向的„[向,,因此
"
U = |U| e
– /0,23K
U xi + U y j + U z k = |U| (e xi + e y j + e z k )
A@0,
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A@0,
& 方向餘ÔK
– 因此K
U x = |U| e x, U y = |U| e y, U z = |U| e z
– ä些°:與° (2.15) 作³較K
c!" θ x = e x,
c!" θ y = e y,
c!" θ z = e z
– U 的方向餘Ô‚於與q同方向!„[向,的0,;
A@0,
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A@0,
& /0,23[\向,K
– ÇÈÊ標 # x A) y A, z A$ 的W A
/TÊ標 # x B) y B, z B$ 的W
B ;
A@0,
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A@0,
& /0,23[\向,K
– f A 到 B 的[\向, r AB K
r AB = ( x B − x A)i + ( y B − y A) j + ( z B − z A)k #2'&!$
A@0,
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A@0,
& 於ñ定Ù!向,的0
,K – &A@7用V,常/與向
, U !Ù8的g個W A 與 B 的Ê標©定,
1定…向, U 的方向
– î用° (2.17) 決定[\向,
r AB
A@0,
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A@0,
j於ñ定Ù!向,的0,K
mä r AB ¤/¹的deÃ到一個fW A ©向W B 的„
[向, e AB;
– e AB 的方向與 U X同
m用 U 的de與
e AB XŸ的ŸB K U = |U| e AB
ª9 2 6
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ª9 2.6
方向餘Ô
椼架的 C W的Ê標“ xC = 4 $) yC = 0) z C = 0 , W的Ê標
“ x D = 2 $) y D = 3 $) z D = 1 $'
f C W到 W的[\向, rCD 的方向餘Ô“
什麼%
ª9 2 6 ( ¶ )
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方±
ê道 C W# 的Ê標,()能ðµ rCD 的0,;然w
()能×. rCD 的de ( f C 到 的hi ) ã且Ë用
° (2.15) .µ的方向餘Ô;
ª9 2.6 ( ¶ )
ª9 2 6 ( ¶ )
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¯·
E向,0,1.µ[\向, rCD ;
rCD = ( x D − xC )i + ( y D
− yC ) j + ( z D − z C )k
= (2 − 4)i + (3 −
0) j +
(1 − 0) k ($)= −2i + 3 j + k ($)
ª9 2.6 ( ¶ )
ª9 2 6 ( ¶ )
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¯·
×. rCD 的de;
( ) ( ) ( )
$3.74
$1$3$2222
222
=++−=
++= z CD yCD xCDCD rrrr
ª9 2.6 ( ¶ )
ª9 2 6 ( ¶ )
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¯·
×.方向餘Ô;
267.0$3.74
$1 c!"
,802.0$3.74
$3 c!"
,535.0
$3.74
$2 c!"
===
===
−=−==
CD
z CD z
CD
yCD
y
CD
xCD x
rr
r
rr
r
θ
θ
θ
ª9 2.6 ( ¶ )
ª9 2 6 ( ¶ )
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»習¼
椼架的W B 的Ê標“ x B = 2.4 $) y B = 0) z B = 3 $ ;®×.f
W B ©向 W!„[向, * BD 的0,;
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Chapter 2 Engineering Mechanics: STATICS
Fifth EditionCopyright 2008 Pearson Education South Asia Pte Ltd
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