Dimension Definition and 906 Threads

In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one (1D) because only one coordinate is needed to specify a point on it – for example, the point at 5 on a number line. A surface such as a plane or the surface of a cylinder or sphere has a dimension of two (2D) because two coordinates are needed to specify a point on it – for example, both a latitude and longitude are required to locate a point on the surface of a sphere. The inside of a cube, a cylinder or a sphere is three-dimensional (3D) because three coordinates are needed to locate a point within these spaces.
In classical mechanics, space and time are different categories and refer to absolute space and time. That conception of the world is a four-dimensional space but not the one that was found necessary to describe electromagnetism. The four dimensions (4D) of spacetime consist of events that are not absolutely defined spatially and temporally, but rather are known relative to the motion of an observer. Minkowski space first approximates the universe without gravity; the pseudo-Riemannian manifolds of general relativity describe spacetime with matter and gravity. 10 dimensions are used to describe superstring theory (6D hyperspace + 4D), 11 dimensions can describe supergravity and M-theory (7D hyperspace + 4D), and the state-space of quantum mechanics is an infinite-dimensional function space.
The concept of dimension is not restricted to physical objects. High-dimensional spaces frequently occur in mathematics and the sciences. They may be parameter spaces or configuration spaces such as in Lagrangian or Hamiltonian mechanics; these are abstract spaces, independent of the physical space we live in.

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  1. V

    Motion in one dimension physics problem

    Homework Statement Determine the stopping distances for an automobile with an initial speed of 90 and human reaction time of 1.0 : (a) for an acceleration = -5.0 , (b) for = -7.5 . Homework Equations vf = vo + at avg velocity = (vf + vo) /2 d = vo)t + (1/2) at2 vf2 = vo2 + 2ad...
  2. V

    Motion in one dimension problem

    Homework Statement An unmarked police car traveling a constant 85 is passed by a speeder. Precisely 1.00 after the speeder passes, the police officer steps on the accelerator. If the police car accelerates uniformly at 3.00 and overtakes the speeder after accelerating for 5.00 , what was the...
  3. N

    Constant Acceleration one dimension

    Homework Statement A rocket starting at rest takes on a net acceleration of 20m/s^2 in a vertical line until it runs out of fuel after 5 seconds. At what height does it run out of fuel? What is its velocity when it runs out of fuel? What is its maximum height? How long does it take to hit the...
  4. A

    How do I calculate the height of a tower using motion in one dimension?

    I have a few problems I'm having trouble with. If I can get some help with this one I should be able to figure out the rest I have. 1. A rock is thrown downward from the top of a tower with an initial speed of 12 m/s. If the rock hits the ground after 2.0 s, what is the height of the tower...
  5. L

    What is the fourth variable in the 4th dimension?

    is it time? what's your theory on the study of other dimensions? and applications from them?
  6. nomadreid

    Fractal dimension of CMBR? Cluster distribution?

    I have read speculations that (1) the cosmic microwave background radiation has a fractal distribution (non-integral Hausdorff dimension), and (2) the same might be true of galaxy cluster distribution (although different dimensionality to (1)) Whether or not one or both analyses are...
  7. J

    Primary Dimension: Mass Units for Power, Pressure, Modulus, Angular Velocity

    Homework Statement For each quantity listed, indicate dimensions using mass as a primary dimension and give typical SI and English units: power pressure modulus of elasticity angular velocity Homework Equations The Attempt at a Solution im not sure i understand what it is...
  8. A

    Hilbert space dimension contradiction

    Hi, I was wondering how the state vector for a particle in a 1-D box can be expanded as a linear combination of the discrete energy eigenkets as well as a linear combination of the continuous position eigenkets. It seems to me that this is a contradiction because one basis is countable whereas...
  9. C

    Ball kinematics in one dimension

    Homework Statement A ball is thrown upward from the top of a 25.0-m-tall building with an initial speed of 12m/s. At the same time, a person is running on the ground at a distance of 31.0m from the building. What is the person's average speed if he catches the ball at the bottom of the...
  10. C

    Comparing Velocities and Accelerations of Two Motorcycles in One Dimension

    Homework Statement Two motorcycles are traveling due east with different velocities. Four seconds later, they have the same velocity. During this 4s interval, motorcycle 1 has an average acceleration of 2.0 m/s(squared) due east, while motorcycle 2 has an average acceleration of 4.0...
  11. L

    Elastic collision in one dimension

    Homework Statement A Pendulum weighs .5kg and has a string length of 70cm swings from a horizontal position downwards to hit a block that weighs 2.5 kg and is on a frictionless plain. Calculate the speed of both the ball and the block after the elastic collision. Homework Equations...
  12. M

    Proving W=V When W is a Subspace of an n-Dimensional Vector Space

    Homework Statement Prove that if W is a subspace of an n-dimensional vector space V and dim(W) = n, then W=V Homework Equations The Attempt at a Solution I don't know where to start.
  13. J

    Linear Algebra: dimension of subspace question

    Homework Statement Find an example of subspaces W1 and W2 in R^3 with dimensions m and n, where m>n>0, such that dim(intersection of W1 and W2)= n Homework Equations dim(W1+W2)= dim(W1) + dim(W2)-dim(intersection of W1 and W2) The Attempt at a Solution Well what I know...
  14. Y

    Acceleration in one dimension

    Homework Statement "2 cars C and D travel in the same direction in straight line. During particular time interval t, car D is AHEAD of car C, and D is speeding up while car C is slowing down. During the interval t, it is observed that the distance between the cars decreases. Explain how this...
  15. H

    Dimension of Angle: Learn LL^-1 from Course Book

    In my course I am curios to know how the dimension of angle becomes [LL^-1]. The following pic is taken by course book. http://i42.tinypic.com/iy2y3r.jpg
  16. K

    Time as Dimension: Before & After Einstein's G.T.R.

    Since when is Time considered to be a dimension? Is it after Einstein's G.T.R. or was it considered to be one before it?
  17. J

    What is the dimension of its kernel?

    Homework Statement Suppose that dim V = m and dim W = n with M>=n . If the linear map A : V -> W is onto, what is the dimension of its kernel? Homework Equations The Attempt at a Solution Onto, means that every vector in W has at least one pre-image therefore, the kernel can...
  18. B

    Subspace & Dimension of V in P2 - Closure and Dimension Analysis

    Homework Statement Is the collection a subspace of the given vector space? If so what is the dimension? V={ax^2+bx+c: a=b+c} in P2 Homework Equations The Attempt at a Solution The first part of the question is pretty straightforward. I just verified closure under addition and...
  19. M

    How Long Does It Take for Two Rocks Thrown from a Bridge to Hit the Water?

    Homework Statement Heather and Jerry are standing on a bridge 46 m above a river. Heather throws a rock straight down with a speed of 30 m/s. Jerry, at exactly the same instant of time, throws a rock straight up with the same speed. Ignore air resistance. How much time elapses between the...
  20. M

    How Far Will a Car Coast Up a Hill After Running Out of Gas?

    Homework Statement A car traveling at 25.43 m/s runs out of gas while traveling up a 17° slope. How far up the hill will it coast before starting to roll back down? Homework Equations Not sure on the equation to be used? The Attempt at a Solution
  21. marcus

    Nice LQG paper by Modesto about dimension down with scale

    This confirms a beautiful result of Modesto's earlier paper, that LQG seems to coincide with two other very different types of quantum geometry/gravity on the business of dimensionality declining continuously with scale. Fractal-like microstructure of space time. As scale decreases the observed...
  22. S

    Electromagnetic force in the fourth spatial dimension?

    Purely from a non-mathematical perspective, could some/any/every-body answer my questions on the electromagnetic force in extra dimensions? άλφα) I am curious about how in the old Kaluza theory of the electromagnetic force the equation, which he found from his equations worked when it...
  23. H

    How long did it take for the box to travel 4.00 meters?

    Homework Statement A box slides along a surface with a positive initial velocity. It the experiences an acceleration of -0.25m/s^2. After traveling 4.00 meters, its velocity is +0.50m/s. How long did it take for the boxto travel the 4.00 meters? Homework Equations \Deltax=Vot + 1/2 a t^2...
  24. Spinnor

    Suppose there is an extra compact dimension, and suppose

    Suppose there is an extra compact spatial dimension in addition to familiar space dimensions x, y, and z. Let us suppose that matter is some kind of 3 dimensional surface moving in these 4 spatial dimensions, 4-space, in some cyclical manner. Let us suppose that from the shape of the surface we...
  25. J

    Is Time Truly a Uni-Dimensional Force?

    Hi there, I have a question. To me, measuring spatial dimensions is quite a bit different than measuring the time dimension. I was thinking the other day and it seemed very obvious to me what the main difference was. Time never goes the other way. A dimension defines two equal and...
  26. P

    Linear Algebra: Null Space and Dimension

    Homework Statement Prove that dim(nullA) = dim(null(AV)) (A is a m x n matrix, V is a n x n matrix and is invertible Homework Equations AX=0 and AVX = 0 Null(AV) = span{X1,..Xd} Null(A) = span{V-1X1,.., V-1Xd} The Attempt at a Solution so you need to prove that...
  27. Z

    Could GR generalized to non-integer dimension?

    could GR generalized to non-integer dimension?? let us suppose that the dimension of space time is NOT an integer then , could we generalize GR to obtain an expressions of Tensor, Covariant derivatives... in arbitrary dimensions ?? let us say 4.567898.. or similar, i mean GR in non integer...
  28. O

    Range of the Hausdorff dimension

    My analysis textbook mentioned in passing that the range of the Hausdorff dimension is all nonnegative real numbers, i.e. for any nonnegative real number a, there's some compact subset of R^n whose Hausdorff dimension is exactly a. The problem is that I don't see how to prove this (and my...
  29. D

    Optimization - Find dimension of a cup that uses least amount of paper

    Homework Statement A cone-shaped paper drinking cup is to be made to hold 30 cm3 of water. Find the height and radius of the cup that will use the smallest amount of paper. Homework Equations volume of a cone (1/3)(pi)(r^2)(h) = 30 SA of a cone pi(r)[sqrt(r^2 + h^2)] The Attempt at...
  30. R

    Dimension and Orthogonality in Vector Spaces: A Proof of the Inequality m ≤ n

    Homework Statement If {u1, u2,...,um} are nonzero pairwise orthogonal vectors of a subspace W of dimension n, prove that m \leq n. The Attempt at a Solution I look at all my notes but I still can't understand what this qurstion asks or what definitions I need to be using for this... I'm...
  31. T

    Reynolds number - Characteristic dimension

    Are there any reference tables for what to use as the characteristic dimension of various shapes when calculating the Reynolds number? I am very interested in ballistics, and have done research and calculations for various ways of estimating the drag coefficient of various bullet shapes. I...
  32. U

    Dimension of preimage question

    let g be the inverse function of f (not necessarily a bijective inverse function) If S is any subset of W then the pre-image of S is g(S) = {v ε V: f(v) ε S} Suppose that U is a subspace of W. Prove that g(U) is a subspace of V. Also prove that dim( g(U) )= dim( U intersect Im(f) ) +...
  33. A

    Vector Space Dimension: Real vs Complex Coefficients

    Homework Statement Let V be a vector space over C of dimenson n . We view V also as a vector space over R by restricting the scalar multiplication of C on V to R .Show that dimR(V) = 2n Homework Equations The Attempt at a Solution I have to show that if x1,...xn form a basis of V...
  34. T

    Reducing the pipe dimension to a heating device

    Let's say there is a water based heating device that has 3/4" connection nipples. The pipe that leads to the device is 5/4". If one ignores pressure- and head loss, does it make any difference how far from the device the pipe reduces from 5/4" to 3/4" ?
  35. A

    What is the null space basis and dimension of A in R^5?

    Homework Statement find a basis of the null space N(A) in R^5 of the matrix A = 1 -2 2 3 -1 -3 6 -1 1 -7 2 -4 5 8 -4 in M3*5 (R) and hence determine the dimension Homework Equations The Attempt at a Solution i found that A= 1 -2 2 3 1 0 0 1/5 2/5 -2/5 0 0 0 0 0 by...
  36. K

    Dimension of a set of symmetric matrices & prove it's a vector space

    Prove: the set of 3x3 symmetric matrices is a vector space and find its dimension. Well in class my prof has done this question, but I still don't quite get it.. Ok, first off, I need to prove that it's a vector space. The easy way is probably to prove that it contains the zero space and...
  37. Mentallic

    Exploring the 4th Dimension of Electromagnetic Waves

    So I've been told that electromagnetic waves oscillate like flicking a rope up and down or a ripple in a pond. In the first example, from a side view, the rope can be considered to be a line, or 1-dimensional. However, once the ropes starts oscillating to represent waves, width must also be...
  38. E

    Divergence theorem in four(or more) dimension

    I don't know whether it was proved or can be prove. I don't know whether it is useful. maybe it can be used in string theory or some other things. any comment or address will be appreciated.
  39. C

    Dimensions Creation: Theory or Always Existed?

    out of curiosity is there any theory on how dimensions that we exist in were created? or are they always said to have existed.
  40. E

    How Do You Calculate the Initial Velocity of a Car in a Perpendicular Collision?

    Homework Statement A motorcycle with a mass of m1 = 200 kg and a speed of v1 = 120 km/h collides with a car of mass m2 = 600 kg, traveling in a perpendicular direction to the motorcycle. After the collision, both vehicles travel in a direction that forms an angle of 60°. Find the initial...
  41. B

    Volume of Dimension reduced due to gravitation

    I've had this on my mind for a while. I'm pretty confident that I have the right idea, but I can't find a formula that I could use to prove it to someone. So, here it is. Imagine a sphere of space-time with absolutely nothing inside of it. It is a large sphere, sun-size large. Now, take...
  42. B

    Is there a formula for the shrinking of the universe due to object curvature?

    Earlier, I had a thought, and I have now been curious about the equation of this thought. Einstein's general relativity obviously states that objects curve and distort space. But the whole reason he came up with a cosmological constant was that he found that the curvature of all of the objects...
  43. C

    Dimension Proof: U + W = dim(U) + dim(W) - dim(U ∩ W)

    Homework Statement Let U, W be subspaces of a vector space V. Show that dim(U+W) = dim(U) + dim (W) - dim (U intersect W). Homework Equations The Attempt at a Solution I can see this picture-wise in a venn-diagram form. In adding U and W you count the elements in their...
  44. Y

    Answer: Lie Algebras: Adjoint Representations of Same Dimension as Basis

    Hello, I hope it's not the wrong forum for my question which is the following: Is there some list of Lie algebras, whose adjoint representations have the same dimension as their basic representation (like, e.g., this is the case for so(3))? How can one find such Lie algebras? Could you...
  45. Ivan Seeking

    HG Wells: The first to call time a fourth dimension?

    The claim was made tonight on a program about science fiction writers that HG Wells was aware of a geometry of 4 dimensions, and then made the leap of faith that time was a 4th dimension - ten years before Einstein came along. Did he really conceive of this before anyone else? I never read...
  46. D

    Solving Dimension Analysis Homework: Q1 & Q2

    Homework Statement http://img410.imageshack.us/img410/8495/88748860vk8.png Homework Equations The Attempt at a Solution I don't know how to deal with the q1 part. q2 = u* =u x sqrt(... so that is clear. Normally I just substitute the expression for u in terms of u* (= f* )and t in terms of...
  47. D

    Dimension Analysis Homework: Buckingham Theorem

    Homework Statement http://img21.imageshack.us/img21/613/70858934fn5.png Homework Equations Buckingham theoremThe Attempt at a Solution My question is what is the difference between question a and b? The sine doesn't influence the dimension. Or is it a question to trick me?
  48. G

    Dimension of U,V and W over K: Do they Equal?

    Just have a question if U,V and W are over the same field K does it mean that dim U = dim V = dim W ?
  49. P

    Understanding Matrix Dimensions: Calculating Entries and Recognizing Symmetry

    How do we calculate the dimension of a matrix? Is it the number of entries? Or is it the number of different entries? For instance if I have a matrix 2x2 the dimension would be 4 but if the matrix is simetrical it would be 3. Is this correct? Thanks for your help.
  50. J

    Dimension of row/ column space

    Homework Statement In the following exercises verify that the row rank is equal to the column rank by explicitly finding the dimensions of the row space and the column space of the given matrix. A = [1 2 1 ; 2 1 -1] Homework Equations The Attempt at a Solution All i can...
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