The proof of mass vector.
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From: Ka-In Yen on 29 Aug 2006 20:27 Danny Dot wrote: > -- > Danny Dot > Look at my site and see how NASA treats a creative mind!!! > The summary is "Not Very Well" :-) > www.mobbinggonemad.org > > > "Ka-In Yen" <yenkain(a)yahoo.com.tw> wrote in message > news:1156469606.545703.194940(a)i42g2000cwa.googlegroups.com... > > > > 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? > > > > I have always thought of mass as a scalar, not a vector. Momentum is a > vector, but not mass. Maybe I am wrong. > Dear Danny Dot, Thank you for your comment. You are right; mass is scalar. In three dimensional vector algebra, linear mass density and surface mass density are vectors; shortly I call them mass vector. The proof of mass vector. Ka-In Yen yenkain(a)yahoo.com.tw 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: Eric Gisse on 31 Aug 2006 15:46 Danny Dot wrote: > -- > Danny Dot > Look at my site and see how NASA treats a creative mind!!! > The summary is "Not Very Well" :-) > www.mobbinggonemad.org How nice of you to put a sig at the beginning and end of your post. Just in case we can't figure out who is talking. > > > "Ka-In Yen" <yenkain(a)yahoo.com.tw> wrote in message > news:1156469606.545703.194940(a)i42g2000cwa.googlegroups.com... > > > > 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? > > > > I have always thought of mass as a scalar, not a vector. Momentum is a > vector, but not mass. Maybe I am wrong. Mass is a scalar. Momentum is a vector. Length is a scalar. Wasn't that easy? > > Danny Dot > www.mobbinggonemad.org
From: Ka-In Yen on 3 Sep 2006 20:46 Ka-In Yen wrote: > Danny Dot wrote: > > -- > > Danny Dot > > Look at my site and see how NASA treats a creative mind!!! > > The summary is "Not Very Well" :-) > > www.mobbinggonemad.org > > > > > > "Ka-In Yen" <yenkain(a)yahoo.com.tw> wrote in message > > news:1156469606.545703.194940(a)i42g2000cwa.googlegroups.com... > > > > > > 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? > > > > > > > I have always thought of mass as a scalar, not a vector. Momentum is a > > vector, but not mass. Maybe I am wrong. > > > Dear Danny Dot, > > Thank you for your comment. You are right; mass is scalar. > In three dimensional vector algebra, linear mass density and > surface mass density are vectors; shortly I call them mass > vector. > Dear Danny Dot, Do you have any comment on the paper "Magnetic force: Combining Drag force and Bernoulli force of ether dynamics"? http://www.geocities.com/redlorikee/mdb2.html If you find any flaws of the theory, please kindly advise me.
From: Ka-In Yen on 1 Oct 2006 22:08 Physicists recall. A fault is found in their logic circuit: they are ill-trained in 3D vector algebra. 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: Eric Gisse on 1 Oct 2006 23:43
Ka-In Yen wrote: > Physicists recall. A fault is found in their logic circuit: they are > ill-trained in > 3D vector algebra. Says the guy who can't use the standard notation to save his life. [snip stupidity] |