Vector

※數碼子

i want to know what "graph" in 3D do they represent, i cannot image ...

Naozumi

第一個
x,y,z軸將個空間分開8份, z>=0,咪即係最上面果四份

第二個
x^2 + y^2 + z^2 係一支vector的長度
咁 x^2 + y^2 + z^2 = 1, 咪即係支vector的長永遠係1,咪即係一個球體個surface
而 x^2 +y^2 + z^2 <=1 即係球體個面加埋interior

第3個
x^2 + y^2 = 1係圓形,但係z係任意,即高度係任意,咁即係好多個圓係唔同高度,所以係冇頂冇底的cylinder

第4個,我俾d提示你啦, 先唔好理 z = 1果部份
當z = 1時,條equation係乜? z = 2又點, z= 3, 4, 5,....又點

[ 本帖最後由 Naozumi 於 2008-9-1 04:50 PM 編輯 ]

※數碼子

今日好灰,個professor問有幾多人識3D vector,成大半班人識,好屈....=(

x^2 + y^2 + z^2 係一支vector的長度　＜－一支vector的長度唔係sqr (x^2 + y^2 + z^2)咩？


4) 考慮左z=sqr(x^2+y^2) 先...
sqr(x^2+y^2)係xy plane vector
當z increase gradually, xy plane vector (圓)沿住z axis ,個radius increase gradually...
去到z=1 停左, so 係一個倒圓錐體...(個面加埋interior)(係最上面果四份)

[ 本帖最後由 ※數碼子 於 2008-9-1 06:14 PM 編輯 ]

Naozumi

哈哈,因為現今的Pmaths課程係cut左 3D vector cogeom,但係有有d老師會私人再教

你咪諗到第4個lor,咪幾好,係初時真係唔適應. 我仲記得我第一份數學功課全部唔識做,真係好灰

c443122

wa...呢d 就係大學數@.@ 歎為觀止 @_@

Naozumi

呢d只係以前中六課程,仲未到大學數戲肉,嘿嘿

※數碼子

戲肉係咩?=.=.......我見real analysis d notes好得人驚......

kwoklap

原帖由 ※數碼子 於 2008-9-1 04:07 PM 發表
i want to know what "graph" in 3D do they represent, i cannot image ...

你呢d禁似我係UST讀緊果D,我份note都有你問緊果part

※數碼子

YES, i study ust math ...but my math sucks...sosad

kingdee1666

請勿作無意義的回覆,特此警告

[ 本帖最後由 Naozumi 於 2008-9-3 10:20 PM 編輯 ]

kingdee1666

請勿作無意義的回覆,特此警告
由於重犯, 根據版規作出懲處

[ 本帖最後由 Naozumi 於 2008-9-3 10:23 PM 編輯 ]

※數碼子

Prove using vector
1.If P, Q,R,S are mid point of AB,BC,CD,DA respectively,of quadrilateral ABCD,then PQRS is a parallelogram.
2.The diagonals of any  parallelogram bisect each other.

Naozumi

2題數都係Add math程度喎,你真係自己度一度先啦
特別第1題只係簡單的divison of vector的應用

第2題,假設 a+b/2 係其中一條diagonal的中點的position vector, 咁你只要計埋另一條diagonal的中點的position vector,一對就知

[ 本帖最後由 Naozumi 於 2008-9-7 12:57 AM 編輯 ]

※數碼子

哈哈,recall 返d memory 之後度到~~~
1.Given r, s, t, r=/=0, s=/=0 and a st | a | ^2>4rst
Solve system of equation rx+sy=a and x.y=t

2.Let a be a non-zero vector.
Describe the set of all point in 3space whose positive vector r satisfy a.r=0

2.計到    a1r1+a2r2+a3r3=0....即係咩geometry meaning?

[ 本帖最後由 ※數碼子 於 2008-9-7 11:51 PM 編輯 ]

Naozumi

第1題未度到

第2題...你真冇學過3D vector cogeometry,真係唔掂喎,佢其實係一個通過origin的plane
至於點解,你自已查下書, 俾小小hints: plane有樣野叫normal,而家a就果支normal

※數碼子

If u.(vxw)=/=0 and X is an arbitrary 3-vector, find the numbers a,b,c s.t.
X=au+bv+cw.

Naozumi

This is not a difficult question, by property of scalar triple product

Xw = auw + bvw
Xw.u = 0 + bvw.u

Then...

※數碼子

1.find the equation of the plane that passing through the lines x+y=2,y-z=3, and perpendicular to the plane 2x+3y+4z=5.

2.find the equation of the plane that through (1,2,-1) and make equal angles with positive direction of coordinate axes.

thanks

[ 本帖最後由 ※數碼子 於 2008-9-25 08:27 AM 編輯 ]

Naozumi

第1題果兩個  x+y = 2同 y - z = 3係plane黎喎....

第2題
用 cos^2 a + cos^2 b + cos^2 c = 1
再用plane equation
x cos a + y cos b + z cos c = constant

※數碼子

i want to ask how to find the normal of plane in expression
r(s,t)=(-3,4,9)+s(9,-5,9)+t(3,-2,3).

and there is one more question.
a. find all unit vector in 3D space that makes an angle of 1 radian with both vectors i+2j+k and 2i-2j+k.
b. let v0 be a vector with its initial point at the origin. all vectors with initial points at the origin that make an angle A with v0 lie on a cone whose axis is along v0. what does this imply about the vectors obtained in (a).

thanks

[ 本帖最後由 ※數碼子 於 2008-9-30 07:11 AM 編輯 ]