DQW

2. The cross-product of a pair of vectors is a vector that is normal to the plane containing the two vectors and whose length is given by |u||v|sin(theta) {which is the area of the parallelogram made by u and v}. If two vectors are parallel, their cross product is a null vector, so it may be useful to use the cross product to check parallelism.

 

The dot product is a scalar whose value is given by |u||v|cos(theta). When two vectors are perpendicular to each other, their dot product os zero, so it is useful to test orthogonality using the dot (or inner) product.

 

You can also use either product to determine the angle between a pair of known vectors, but since the dot product is easier to calculate, one tends to use that.

1. http://mathworld.wolfram.com/VectorSpaceProjection.html

 

so, cross with orthogonal and dot with proportional?

What does that mean ?

 

You do not use the cross production at all in finding the projection of u onto v. The projection is given by

 

[math]proj (\vec{u} ~on~ \vec{v}) = |\vec{u}| cos(\theta) \hat{v} [/math]

where [imath]\hat{v}[/imath] is the unit vector along [imath]\vec{v}[/imath]. So, it can also be written as :

 

[math]proj (\vec{u} ~on~ \vec{v}) = \frac{(\vec{u} \cdot \vec{v})}{|\vec{v}|} \hat{v} = \frac {(\vec{u} \cdot \vec{v})}{|\vec{v}|} ~\frac{\vec{v}}{|\vec{v}|}[/math]

 

If you've done Newtonian Mechanics, this is exactly what you do when you resolve vectors (forces) along a pair of orthogonal directions. You find the projections of those vectors onto the required pair of orthogonal unit vectors.

I have no idea why no need the hall effect to make a capacitor. A capacitor is merely some arrangement of conducting plates. They are charged quite trivially using a battery and a switch. The hall effect does nothing to help you make/charge a capacitor.

One of the first things we learn in science is that matter can neither be created nor destroyed. But what about when matter and antimatter collide? I've been reading "The Elegant Universe"; and it talks about matter and antimatter colliding and destroying each other. So how does this all fit together? Has that law since been changed?

There was a Law of Conservation of matter/mass, which, after Einstein must be modified (to incorporate mass-energy conversions) such that it becomes identical to the Law of Conservation of Energy (which is now the relativistic energy and includes a rest energy term, [imath]m_0c^2[/imath]).

 

When matter and anitimatter annihilate each other, they emit photons of total energy equal to the total relativistic energy of the colliding particles. In other words, the mass of the particles gets converted into the energy of the emitted photons.

 

Also, exercise some caution when reading "EU".

I can't say I understood the Atom / Molecule explanation - only in that the size of an 'grape' atom is to small in relation to a Molecule - so in essence that was what i really to clear up. I accept that Atoms cannot just be 'sized/measured' without looking at the other surroundings.

A molecule is just a bunch of atoms "stuck" to each other. An oxygen molecule (represented as O₂) has two atoms of oxygen, and so, is no more than twice as big as an O atom. On the other hand, there are some truly large molecules (like proteins) which are made up of hundreds or thousands of atoms, and hence are much (tens or hundreds of times) bigger than one of those atoms.

 

As for measuring the size of atoms, that has been done pretty well too.

Also, you should look into generalized coordinates. The idea of using properties like position and (generalized) momentum as coordinates in some phase space is not new. It's been around since the 18th century (Lagrange, Hamilton, and then Poincare').

I suggest you look into Bell's Theorem and the EPR Paradox.

My questions are:

1) In 3-space' date=' what is the locus of points equally distant from a point and a plane? (My guess is a paraboloid.)[/quote']Correct.

 

2) In 3-space, what is the locus of points equally distant from a line and a plane?

I think it is a pair of congruent, coaxial, oblique cones with vertex at the point of intersection.

 

When the line is parallel to the plane, it is clearly the surface of translation (parallel to the line) of a parabola.

 

3) In 3-space, what is the locus of points equally distant from two lines?

In general, for a pair of skew lines, I guess this surface would be something like a distorted saddle.

 

For a pair of parallel (coplanar) lines, the locus would be the plane normal to the plane containing the lines and midway between them.

 

For a pair of intersecting, coplanar lines, the loci would be the pair of planes normal to the plane that contains them, and hwose intersections with this plane are the angle bisectors of the lines.

But aren't typical output impedances of the order of some milliohms ? How will that sufficiently serve as a current limiter ?

You are being asked to design a rudimentary version of a UV/Vis (from the suggestion of an OHP) spectrometer.

 

http://www.ent.ohiou.edu/che/che415/UV_w02.html

How do you get

 

[math]\frac{|x - 4|}{|\sqrt(x) +2|} ] <\delta\frac{1}{\sqrt(5) + 2}[/math] ?

Without doing any math, less than a billionth of a second. So a (very small) fraction of a billionth of a second.

Where did you get this from ? From the expectation value of the distance to the first air molecule ?

QED is an abbreviation of Quod Erat Demonstrandum, which is Latin for "...which was to be demonstrated."

 

Shall look into the problem later today.

2. Use the properties of inner products (along with the definition of orthogonality) and expand the LHS. The RHS will follow in just a few steps.

 

additivity : <u,v+w> <u,v> + <u,w>

 

scaling : <au,v> = a <u,v>

 

conjugation : <u,v> = <v,u>*

1. What do you know about the roots of the minimum polynomial (and the roots of the characteristic polynomial) ?

Okay, don't believe me....

 

http://mathworld.wolfram.com/Binary.html

 

Therefore, 1/2 would be represented as 0.1, 1/4 as 0.01, 3/4 as 0.11, and so on.

unless you use an operator on it, which is in fact NOT expressing it in binary, but as an equation.

Separating terms by an operator creates an expression, not an equation.

Theres no way to express a half in binary

1/2 is expressed as 0.1 in binary.

Is it just me or does anyone else fail to see the signature on the picture?

Perhaps you missed this little gem :

The signature is right there on the photo, but for some strange reason it won't show up when the photo is scanned. Bid only if you believe.

i take that as an, everything is perfectly correct??

Perhaps you (kedas) are neglecting the fact that the wire is capable of stretching elastically to accomodate the amplitude ?

as i stated before, i am pretty sure that operations on infinity are undefined

...they are only undefined in the field of real numbers because infinite numbers are not members of the set of reals.

This thread should be in Politics, or some such place. Terrorist bombings are not independent events, and even modeling them as such does not give rise to a reduced probability of repetition. So, what is being discussed here are the governmental and sociological responses to a terror event, and their likely effects on a repeat attempt.

Just one really tiny correction :

[math]

f_n=\frac{v}{\lambda _n}=\frac{n\sqrt(\frac{F}{\mu})}{2L}=\frac{n\sqrt(F)}{2L\sqrt(\mu)}

[/math]

 

I added a subscript to the frequency to indicate which mode it is for. The rest is all good.