vector

If @i+3j and 3i+(8+@)j are two parallel vector , find the possible value of @.

ABCD is a quadrilateral. P,Q,R and S are mid-point of AB,BC,CD and DA respectively.Prove that PQRS is a parallelogram?

蝎杰

Hust 于 23-2-2005 09:42 PM  说 :
If @i+3j and 3i+(8+@)j are two parallel vector , find the possible value of @.

@i+3j=k{3i+(8+@)j}   because they are parallel vector
@i+3j=3ki+(8k+@k)j
@=3k 3=8k+@k
k=@/3
3=8(@/3)+@(@/3)
9=8@+@^2
@^2+8@-9=0
(@+9)(@-1)=0
@=-9 or @=1

蝎杰

Hust 于 23-2-2005 09:45 PM  说 :
ABCD is a quadrilateral. P,Q,R and S are mid-point of AB,BC,CD and DA respectively.Prove that PQRS is a parallelogram?

我不会post diagram 出来，你画了图画才看解答吧！
let the position vector of A,B,C,D be a,b,c,d
i.e:OA=a,OB=b,OC=c,OD=d
S is a mid point of AD
OP={OA+OD}/2
  =1/2(a+d)
P is a mid point of AB
OP=(OA+OB)/2
  =1/2(a+b)
Q is a mid point of BC
OQ=(OB+OC)/2
  =1/2(b+c)
R is a mid point of CD
OR=(OC+OD)/2
  =1/2(c+d)
SP=SO+OP
  =-1/2(a+d)+1/2(a+b)
  =1/2(b-d)
RQ=RO+OQ
  =-1/2(c+d)+1/2(b+c)
  =1/2(b-d)
vectorSP=vectorRQ
SP=RQ
SP parallel RQ
therefore,SPRQ is a parallelgram

我有数学的疑惑
Using the vector method , prove that the perpendicular bisectors of the sides of a triangle are convergent.

Prove that the diagonals of a rhombus are perpendicular?

蝎杰

Hust 于 25-2-2005 09:13 PM  说 :
Prove that the diagonals of a rhombus are perpendicular?

let OABC is a rhombus with dot OA=a,dot OB=b
from triangle OAB,dot AB=b-a
from triangle OBC,dotOC=a+b
hence,dot(AB.OC)=(b-a)(a+b)
                =b.a+b.b-a.a-a.b
                =|b|^2-|a|^2
                =0  because |a|=|b| in a rhombus

Find the probability of throwing three sixes twice in five throws of six dice?

Leong13

A,B,c are the vertices of a triangle.The perpendicular line from A to BC and the perpendicular line from B to AC meet at D.Show that CD is perpendicular to AB.

fritlizt

Leong13 于 27-2-2005 07:39 PM  说 :
A,B,c are the vertices of a triangle.The perpendicular line from A to BC and the perpendicular line from B to AC meet at D.Show that CD is perpendicular to AB.

∠AGD =  ∠DFB = 90
∠ADG =∠FDB
所以 ∠GAD =∠ DBF
给一个四方形， 内角的合为360度。
所以∠GAE+ ∠GDE = 180
∠GDE= ∠CDB
所以 ∠GAE=∠GAD +∠DAE  
                =∠DCF+∠DBF

∠GAD =∠ DBF,
所以 ∠DAE =∠ DCF.

---> ∠DAE =∠ DCF.∠ADE =∠CDF,
SO,∠AED =∠ DFC = 90.
CD is perpendicular to AB



[ Last edited by fritlizt on 28-2-2005 at 02:13 PM ]

Leong13

数学的疑惑
1.An aircraft that has a speed of 350kmh-in still air is heading south.The velocity of the wind from the north-east direction is 50kmh- .Determine the resultant velocity of thr aircrafy.
2.A force of 7N and a force P has a resultant force of magnitude 15N.Find P if the angle between the two forces is
(a)90sudut      (b)60sudut.

cristiano~7

Hust 于 27-2-2005 10:35 AM  说 :
Find the probability of throwing three sixes twice in five throws of six dice?

6個中出現3個6的機率

(A,B,C,D,E,F) 中
A 代表第一個出現的數字
B 代表第二個出現的數字
C 代表第三個出現的數字
D 代表第四個出現的數字
E 代表第五個出現的數字
F 代表第六個出現的數字

3個6可以有以下的組合 :
3個6和其他三個數字組成
3個6和其他兩個數字組成
3個6和其他一個數字組成

(6,6,6,A,B,C)的組合可能 10
(6,6,6,A,B,C)的排列可能 120
(6,6,6,A,B,C)共形成1200種三個6和三個不同的數字組合的可能
解說：
A,B,C 可以有 10 種組合 (5C3)
(6,6,6,A,B,C) 可以有 120 種排列 (6P6 / 3P3)
每個不同的組合，各自有120種排列

(6,6,6,A,A,B)的組合可能 20
(6,6,6,A,A,B)的排列可能 60
(6,6,6,A,A,B)共形成1200種三個6和兩個不同的數字組合的可能
解說：
A,B 可以有 20 種組合 (5C1) * (4C1)
先選出一個數字A，再選出另外一個數字B
(6,6,6,A,A,B) 可以有 60 種排列 (6P6 / 3P3 / 2P2)
每個不同的組合，各自有60種排列

(6,6,6,A,A,A)的組合可能 5
(6,6,6,A,A,A)的排列可能 20
(6,6,6,A,A,A)共形成100種三個6和一個不同的數字組合的可能
解說：
A 可以有 5種組合
(6,6,6,A,A,A) 可以有 20 種排列 (6P6 / 3P3 / 3P3)
每個不同的組合，各自有20種排列

三個六的可能 = 1200 + 1200 + 100 = 2500種

六個數的可能 = 6 x 6 x 6 x 6 x 6 x 6 = 46656種

六個數字中出現三個六的機率 = 2500 / 46656

用隨機程式演算過，46656次中出現3個6的次數為 2500左右
所以答案絕對正確