The proof of mass vector.
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From: Pmb on 5 Feb 2007 13:01 "Ka-In Yen" <yenkain(a)yahoo.com.tw> wrote in message news:1170636619.052197.255940(a)l53g2000cwa.googlegroups.com... > On Jan 30, 9:00 am, "Pmb" <som...(a)somewhere.net> wrote: >> "Ka-In Yen" <yenk...(a)yahoo.com.tw> wrote in message >> >> news:1170117823.060160.318280(a)h3g2000cwc.googlegroups.com... >> >> >> >> > On Jan 29, 9:01 am, "Pmb" <som...(a)somewhere.net> wrote: >> >> "Ka-In Yen" <yenk...(a)yahoo.com.tw> wrote in >> >> messagenews:1170030684.922322.140380(a)a75g2000cwd.googlegroups.com... >> >> > Thank you for your comment. Inverse of a vector is widely >> >> > accepted by mathmaticians and physicists; Clifford proved >> >> > 1/A = A/A^2 (where A is a vector). >> >> > According to the above equation, we have >> >> > k/<a,b,c>=[k<a,b,c>]/[<a,b,c>^2] >> >> I didn't say that the inverse of a vector didn't exist. I said it >> >> wasn't >> >> a >> >> vector. >> >> > Dear Pete, >> > Everytime you find 1/A in a equation, ..... >> >> I'd have to look up its meaning. What does this1/i mean? >> >> > .....you can replace it with >> > A / A^2. Assuming A = a<u>, where <u> is a unit vector, we >> > have 1/<u> = <u> / <u>^2 = <u>. This equation tells that the >> > direction of a vector, unit vector, is not changed by "division". >> > In 3D vector algebra, the direction of a vector is changed by -1 >> > only. >> > 1/A = 1/ (a<u>) = (1/a) <u> >> > 1/A includes two informations: 1) magnitude 2) direction; so >> > it's a vector. >> >> More later > > Dear Pete, > > Do you have any further questions? Ya got me! I totally forgot the entire discussion. I've been under the weather for quite some time. Pete
From: Ka-In Yen on 2 Mar 2007 20:06 Is it useful? The proof of mass vector. Ka-In Yen http://www.geocities.com/redlorikee Introduction: In this paper, we will prove that linear mass density and surface mass density are vector, and the application of mass vector is presented. 1. The unit of vector. In physics, The unit of three-dimensional cartesian coordinate systems is meter. In this paper, a point of 3-D coordinate system is written as (p1,p2,p3) m, or (p:3) m and a vector is written as <a,b,c> m, or <a:3> m or l m<i,j,k> = <a,b,c> m where l=abs(sqrt(a^2+b^2+c^2)) is the magnitude of the vector, and <i,j,k> is a unit vector which gives the direction of the vector. For three reasons, a magnitude of a vector can not add to a scalar: i) The magnitude belongs to the set of vector; it's a portion of a vector. Scalar belongs to a field. ii) The magnitude is real non-negative number, but scalar is real number. iii) The unit of magnitude is meter, but scalar has no unit. This is a major difference between physics and mathematics. 5m+3 is meaningless. 2. Linear mass density is a vector. The mass of a string is M kg, and the length of the string is l m<i:3>. Where l m is the magnitude of the length, and <i:3> is a 3-D unit vector which gives the direction of the string. Then the linear mass density of the string is: M/(l<i:3>)=(M/l) (kg/m)<i:3> The direction, <i:3>, is not changed by "division", so we can move <i:3> from denominator to numerator. A direction is changed by -1 only. A proof is found in Clifford algebras: [Proof] k/<a,b,c>=[k<a,b,c>]/[<a,b,c>^2] =(k/l) <i,j,k> where l is the magnitude of <a,b,c>, and <i,j,k> is the unit vector of <a,b,c>. [Proof] 3. Surface mass density is a vector. A parallelogram has two vectors: l m<i:3> and h m<j:3>. <i:3> and <j:3> are unit vectors. The area vector of the parallelogram is the cross product of these two vectors. l m<i:3> X h m<j:3>= lh (m^2 )<i:3>X<j:3> = lh abs(sin(theta)) (m^2)<k:3> Where theta is the angle between <i:3> and <j:3>. <k:3> is a unit vector which is perpendicular to <i:3> and <j:3>. For AXB=-BXA, an area has two directions. We can divide the area vector by the length vector. lh*abs(sin(theta))<k:3>/[l<i:3>] =h<i:3>X<j:3>/<i:3> =h(<i:3>X<j:3>)X<i:3> (The direction, <i:3>, is not changed by "division", and the division is replaced by a cross product.) =-h<i:3>X(<i:3>X<j:3>) =-h[<i:3>(<i:3>o<j:3>)-<j:3>(<i:3>o<i:3>)] (where o is dot product.) =-h(cos(theta)<i:3>-<j:3>) =h(<j:3>-cos(theta)<i:3>) m The result is a rectangle, not the original parallelogram. We can test the result. h(<j:3>-cos(theta)<i:3>)Xl<i:3>=lh m^2<j:3>X<i:3> The magnitude of the area vector is conserved, but the direction is opposite. The mass of a round plate is M kg, and the area vector is A m^2<i:3>; then the surface mass density is M kg/(A m^2<i:3>)=M/A (kg/m^2)<i:3> 4. Mass vector in physics. Mass vector has been found in two equations: 1) the velocity equation of string. 2) Bernoulli's equation. i) For waves on a string, we have the velocity equation: v=sqrt(tau/mu). v is velocity of wave, tau is tension applying to string, and mu is linear mass density of string. We can rewrite the equation: mu=tau/v^2. In the above equation, the mu is parallel to tau, and both of them are vector. ii) Bernoulli's equation is: P + k*v^2/2=C (P is pressure, k is volume density, and v is velocity. Here we neglect the gravitational term.) Multiplying cross area vector A m^2<i:3> of a string to Bernoulli's equation(where <i:3> is a unit vector), P*A<i:3> + k*A<i:3>*v^2/2=C*A<i:3> F<i:3> + L<i:3>*v^2/2=C*A<i:3> (where F is the magnitude of force, and L is the magnitude of linear mass density.) These two equations are well used in the theory "Magnetic force: Combining Drag force and Bernoulli force of ether dynamics." For detail, please refer to my site: http://www.geocities.com/redlorikee
From: Androcles on 2 Mar 2007 20:25 "Ka-In Yen" <yenkain(a)yahoo.com.tw> wrote in message news:1172884002.101304.55620(a)z35g2000cwz.googlegroups.com... > Is it useful? > > > The proof of mass vector. The disproof: http://mathworld.wolfram.com/VectorSpace.html When you find negative mass you can call it a vector. Ergo, you are bonkers.
From: Eric Gisse on 2 Mar 2007 20:58 On Mar 2, 4:06 pm, "Ka-In Yen" <yenk...(a)yahoo.com.tw> wrote: > Is it useful? No, because it is wrong. [snip junk]
From: Ka-In Yen on 3 Mar 2007 21:08
On Mar 3, 9:58 am, "Eric Gisse" <jowr...(a)gmail.com> wrote: Did you finish your homework? Home work for Eric Gisse: A rectangle sits in 3D space. The area vector of the rectangle is A, and the legth vector of one side of the rectangle is L. Please find the length vector of the other side of the rectangle? |