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I would like a definition of vector.


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Initial post: Aug 2, 2012 7:44:01 AM PDT
Many books do not do much more than stating vector is all that has magnitude and direction, drawing lines and writing mathematical formulas, I think that their definitions are not clear and would like a definition expressed in words

Posted on Aug 2, 2012 7:47:05 AM PDT
Last edited by the author on Aug 2, 2012 7:49:17 AM PDT
Lj3d says:
http://en.wikipedia.org/wiki/Vector

As you can see, its not just a simple explanation expressed in words which is common in the world of science and engineering so its best to get used to it.

In reply to an earlier post on Aug 2, 2012 7:49:56 AM PDT
Vectors appear in many contexts. Depending on your focus, you may want to get different explanations. Are you primarily interested in math, physics, computing, or biology?

In reply to an earlier post on Aug 2, 2012 7:52:45 AM PDT
I am primarily interested in physics.

Posted on Aug 2, 2012 9:29:18 AM PDT
Brian Curtis says:
This is as good a basic definition as any.

http://physics.about.com/od/glossary/g/vector.htm

'A vector is a mathematical quantity that has both a magnitude and direction. It is often represented in variable form in boldface with an arrow above it. Many quantities in physics are vector quantities.

'A unit vector is a vector with a magnitude of 1 and is often denoted in boldface with a carat (^) above the variable.'

The most familiar everyday example would be a weather report on wind: "Today's wind is averaging 15 mph from the northwest."

In reply to an earlier post on Aug 2, 2012 11:01:30 AM PDT
Last edited by the author on Aug 2, 2012 11:03:09 AM PDT
In basic physics, a primary use of vectors is to add up all the various forces that might be acting on an object.

For instance, a car is subject to the force of gravity, the push of the ground on the tires, the propulsion of the engine/transmission, the friction of the air, and so on. We can see that gravity points down, the ground pushes up, propulsion points forward, and air resistance points backward. The ground exactly offsets gravity, but the air resistance doesn't usually balance out with the force of the engine, so the car moves forward, net. If there was no ground underneath the car (i.e., if it drove off a cliff), then the gravity would take over, and the car would accelerate downward, for a time--until the ground at the bottom of the cliff reasserted itself.

An important distinction is between vector quantities and scalar quantities. For instance, the police are primarily interested in your speed (a scalar), when they give you a speeding ticket. You, on the other hand, are interested not only in your speed, but also your direction, since you don't care how fast you're going if you aren't going to the right place. Now, since you need to know both your speed & your direction, it may be convenient to express this as one quantity (a vector). You can use an arrow to do this, or you could also just give it as an (x,y) point on the cartesian plane. Noting that a point (x,y pair) is at some distance from the origin, we can see that each point also defines a unique vector, by connecting the origin to the point with an arrow. (Mathematically, there is no distinction.)

This works no matter how many dimensions you're in, just by adding more values to make an ordered set of n points. For 3-space, we'd have (x,y,z), but there is no limit to how many numbers we can include in our 'tuplet': (x,y,z,...,n). In this way, we can describe vectors for spaces that have arbitrary numbers of parameters. For example, I could assign a vector to each point (x,y) on my computer screen, and also include and R, a G, and a B value to my parameter space for specifying the color of the pixel, in 3 more dimensions. Then my 'space' would be 5-D: (x,y,R,G,B) I could use this 'space' to completely specify the instantaneous state of my monitor.

Tell me if any of this is helpful, and I will continue, time permitting.

Posted on Aug 2, 2012 11:12:23 AM PDT
Last edited by the author on Aug 2, 2012 11:17:49 AM PDT
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In reply to an earlier post on Aug 2, 2012 11:43:01 AM PDT
Last edited by the author on Aug 2, 2012 11:53:00 AM PDT
A vector is an arrow. It has length (magnitude) and direction.

Vectors are mathematical objects which can be added, subtracted, and multiplied. There are two different kinds of multiplication operations for vectors, the dot product and the cross product.

Vectors can be used to represent certain physical quantities which have magnitude and direction, such as distance, velocity, acceleration, force, momentum, and angular momentum.

Using vectors can make the mathematics of mechanics and other branches of physics much more convenient. For example, supposing you threw a football downfield. It spins around its long axis, but also wobbles as it does so. You can represent the spin around the axis as one vector, and the wobble as another. Then you can easily find the direction and magnitude of the overall spin by adding the two vectors.

You can do calculus with vectors. For example, you can define the derivative of a vector, which is another vector.

In reply to an earlier post on Aug 2, 2012 1:05:18 PM PDT
Re OP: A vector is indeed anything which has both a magnitude and direction. Many things can be described by vectors: motion, force, electrostatic and magnetic fields all qualify. The motion of water in a river can be described by a vector at any point in the water, and these will vary depending on the depth of the river, how close you are to the shore or the bottom, whether there are rocks that produce eddies, and the oars and wake of a rowboat.

If all points of a space have some property that can be described by a vector, one can say that this is a vector space. Various mathematical properties obtain in a vector space; among these are:
- Divergence, meaning that the vectors from nearby points are pointing somewhat away from each other.
- Curl, meaning a tendency of vectors from nearby points to describe a circular pattern around some point.
This is a general picture of the landscape. You can fill in the details by taking a course in vector calculus.

In reply to an earlier post on Aug 2, 2012 2:08:56 PM PDT
Last edited by the author on Aug 2, 2012 2:17:45 PM PDT
Haynes,

If the mathematical quantity of your brain cells wasn't infinitesimal, you'd know that "mathematical quantity", in this context, means "quantity used in mathematical contexts", as opposed to just general discussion.

But since you are a buffoon, I expect you won't get this point, either. If you did, you'd apologize for wasting everybody's time with your annoying drollery, and add something substantive to a discussion. But since you don't know either how to "discuss" or be substantive, you just post stupid garbage, and wait for everyone to vote you down... And THEN you miss the point of that, as well! I imagine, in your foolish conceit, you dream that all those 'no' votes are because you are so much cleverer than all the people you are attempting to needle, who are too dull-witted to recognize you in all your glorious cleverness. I imagine you think you are coming off as "above the fray".

But the reality is, all those 'no' votes simply mean we all wish you'd just go away, and stop wasting everybody's time.

You aren't even fun to argue with, since you don't argue. You aren't even fun to gloat over, since your mind is impervious to outside opinion (and facts, as well.) You have no raison d'etre. You are simply an unpleasant noise in the system.

Among other things, you are giving your religion a bad name. Ask yourself how often J.C. would intentionally condescend to his audience, in a vain attempt to make people feel stupid. If you manage to find an example of said tactics, perhaps in your next post you could use it to justify your existence.

(AHHH! Sometimes it feels good to rant, eh? On the other hand, so does actual thinking. You should give it a try, sometimes, just for a change of pace.)

PS.
Even if you were right, don't you understand that these people here are Samaritans, trying to help someone who has called out for assistance? Is your methodology the proper Christian response to people who have bothered to take a few minutes out of their day to type a note? Does it really matter that maybe they typed an extra word in haste? What sort of man are you, anyway? I mean, really, what is wrong with you? (I was under the impression you styled yourself as a Christian. Well, I guess that was 'my bad', as they say...)

In reply to an earlier post on Aug 2, 2012 2:11:53 PM PDT
Last edited by the author on Aug 2, 2012 2:16:12 PM PDT
a vector is an ordered pair of numbers
written as (first nr., second nr.)
eg (4 , 3 )

nothing more and nothing less

how you interpret that vector depends
a common way is to plot it in cartesian coordinates
the vector would go from origin 0,0 with direction to the ordered pair eg 4,3
and would have magnitude 5 (equivalent to length in your interpretation)

a two valued vector can also be mapped to the imaginary numbers by considering the second value to be the i component

vectors can have many numbers as long as there are at least 2 of them
eg (nr 1, second nr. , 3rd, 4th , 5th)
eg ( 42 , 7, 2.71828 , 3.14159 , 19 )

In reply to an earlier post on Aug 2, 2012 2:28:41 PM PDT
Last edited by the author on Aug 2, 2012 2:30:39 PM PDT
Not to be picky, but your post contradicts itself, and makes at least one false claim.

You say, "a vector is an ordered pair of numbers ... nothing more and nothing less"

and then you go on to tell us how it could be more numbers, which sounds like more, to me.

Secondly, one number is enough. (+6) is a vector on the real line, R1. It has a magnitude (6) and direction (+), and (-6) points in the opposite direction, with the same magnitude.

PS. I await my obligatory no vote.

In reply to an earlier post on Aug 2, 2012 2:29:33 PM PDT
okay
picky picky
you got me
this is not a math class exam
so close enough for govt work i hope

In reply to an earlier post on Aug 2, 2012 2:33:28 PM PDT
I didn't have to wait long, did I?

In reply to an earlier post on Aug 2, 2012 4:01:31 PM PDT
Last edited by the author on Aug 2, 2012 4:03:08 PM PDT
you can write a vector as an ordered pair of numbers, but that doesn't define what the vector means.

In order to do that, you need to define the axes in some manner. Not being a pure mathematician, I'm not sure the minimum that is required in a general case. There should be definition of inner product, as well as some rules for uniqueness (i.e. the axes are not identical to each other, and cannot be interchanged). Orthogonality against the inner product is not required.

In reply to an earlier post on Aug 2, 2012 4:04:40 PM PDT
mathematically that is not so

you could do that in physics or engineering
but it is not a general definition just one for a special case

In reply to an earlier post on Aug 2, 2012 4:13:55 PM PDT
Last edited by the author on Aug 2, 2012 4:17:46 PM PDT
I don't understand which part of my reply you don't like. Without the properties I mentioned, a vector has no meaning. A vector has a direction and magnitude. Without an inner product, there can be no magnitude. Without a definition of uniqueness for the axes, there can be no definition of equality or inequality.

You yourself described the "axes" as possibly being a complex plane. The complex plane has certain properties that allows you to describe an inner product. That inner product provides a magnitude, and also defines the axes as being orthogonal and of unit length. Without that, two numbers are just two numbers. Can you find me a definition online that supports your claim that two numbers by themselves (or one as Randall points out) constitute a vector? Even a number line needs the inner product definition (which is trivial, but cannot be zero for all numbers).

In reply to an earlier post on Aug 3, 2012 12:10:43 AM PDT
Lj3d says:
Government work is good enough for government workers lol, but not for scientific investigation.

In reply to an earlier post on Aug 3, 2012 12:12:36 AM PDT
Last edited by the author on Aug 3, 2012 12:16:00 AM PDT
Lj3d says:
Just once, can you post something without all the creationist refs? We get that your a creationist! No need to be condescending about it. I know you don't wander why others here consider, you a bufoon including me...but maybe you ought to. I don't tell everyone I am an agnostic except where its appropriate to the discussion, and even then I don't act like I'm somehow better because of it. Nor do most of the others here.

In reply to an earlier post on Aug 3, 2012 6:25:33 AM PDT
Brian Curtis says:
Still running and hiding from the question you're scared to answer, eh Haynes? Too bad it's not going to work. Here it is again:

"So basically, every sighting of a UFO, dragon, ghost, Superman, sharks with lasers, or any other outrageous claim made by someone actually DID happen because, after all, they observed it happening, right?"

Posted on Aug 3, 2012 12:59:51 PM PDT
Jack Shandy says:
A vector is an element of a vector space. A vector space is, loosely, a set of objects which can be added to obtain another element of that set, and be multiplied by a real or complex number to obtain another element of that set.

Now, every vector space has a basis, which means that every vector can be written as a sum of scalar multiples of a given set of basis vectors. So, given a basis, any vector in that vector space can be represented by a collection of numbers, which are the coefficients of the basis vectors in the expansion of that vector.

The above is quite abstract, and has nothing to do with directions per se. The set of real numbers is a vector space. The set of square-integrable functions on [0,1] is a vector space. The set of all n-tuples R^n is a vector space.

On differential manifolds, the set of directional derivative operators at any point form a vector space: the tangent space. So to answer the question 'what is a vector' geometrically, I'd say a vector is a directional derivative operator. More abstractly, a vector is just any element of a vector space.

In reply to an earlier post on Aug 3, 2012 1:16:32 PM PDT
Last edited by the author on Aug 3, 2012 1:25:28 PM PDT
Nice post.

I realized that my previous post is incorrect. A vector space does not have to have an inner product defined for it, although it will in many situations (such as any basis that forms a Hilbert Space, or a subset of it).

However it is perfectly fine to have as a basis [apples, water, dollars, love]. There may be no way to define a single scalar magnitude of such a thing, which means no inner product. However, one could still define other properties, such as a degree of happiness.

In reply to an earlier post on Aug 3, 2012 1:22:38 PM PDT
Last edited by the author on Aug 3, 2012 2:14:41 PM PDT
Jack Shandy says:
-"Without an inner product, there can be no magnitude."

Only normed linear spaces where the norm satisfies the parallellogram law are inner product spaces, by which I mean that this specific norm defines an inner product. But there exist norms which do not define an inner product (don't ask for an example though: google is your friend) because they don't satisfy the parallellogram law.

In reply to an earlier post on Aug 3, 2012 1:23:02 PM PDT
Last edited by the author on Aug 3, 2012 1:41:03 PM PDT
Jack Shandy says:
Ninja'd (more or less)

In reply to an earlier post on Aug 4, 2012 4:06:40 PM PDT
Re Lj3d 8-3 12:12 AM: The only reasonable tactic to deal with nitwits such as CH is to put him on Ignore. Which I did, months ago -- to the considerable improvement of my digestion.
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Discussion in:  Science forum
Participants:  11
Total posts:  29
Initial post:  Aug 2, 2012
Latest post:  Aug 6, 2012

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