What is Three dimension: Definition and 13 Discussions
Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called parameters) are required to determine the position of an element (i.e., point). This is the informal meaning of the term dimension.
In mathematics, a sequence of n numbers can be understood as a location in n-dimensional space. When n = 3, the set of all such locations is called three-dimensional Euclidean space (or simply Euclidean space when the context is clear). It is commonly represented by the symbol ℝ3. This serves as a three-parameter model of the physical universe (that is, the spatial part, without considering time), in which all known matter exists. While this space remains the most compelling and useful way to model the world as it is experienced, it is only one example of a large variety of spaces in three dimensions called 3-manifolds. In this classical example, when the three values refer to measurements in different directions (coordinates), any three directions can be chosen, provided that vectors in these directions do not all lie in the same 2-space (plane). Furthermore, in this case, these three values can be labeled by any combination of three chosen from the terms width, height, depth, and length.
So I tried to solve it this way:
The equations of the lines in vector form are
$$(x-2)\hat i+(y-3)\hat j+(z-4)\hat k=\lambda (\hat i+\hat j-K\hat k)$$
$$(x-1)\hat i+(y-4)\hat j+(z-5)\hat k=\mu (K\hat i+2\hat j+1\hat k)$$
Since the lines are some real multiple of the vectors,
For coplanarity...
Firstly, since there is no condition for the z axis in the definition of the potential can I assume that V(x,y,z) = .5mw^2z^2 when 0<x<a, 0<y<a AND -inf<z<inf?
If so then drawing the potential I can see that the particle is trapped within a box with infinite height (if z is the...
Homework Statement
Find the center of mass of a homogeneous solid cube with side ##L## analytically.
Homework Equations
None.
The Attempt at a Solution
I don't understand how to find the center of mass on three dimensions. I know that since it is homogeneous, if I center the cube on the...
Suppose if we have a cube:
The volume of the cube is the product of the length, width and the height. All this time, I've been looking at it as: To get the volume, multiply the area of the cross section of the cube by how many "layers" it has. To elaborate with the diagram given, one can see...
Homework Statement
The direction of vectors A and B are given below for several cases. For each case, state the direction of A X B.
a) A points east, B points south.
b) A points east, B points straight down.
c) A points straight up, B points north.
d) A points straight up, B points straight...
I have function1: x = n(cos((pi/2)-2pi/n))
and function2: y = n(sin((pi/2)-2pi/n))
my goal is to plot a graph where for the same value of n, the x and y are respectively the horizontal and vertical component of the point, this graph should preferably possible to create on a computer or a...
Homework Statement
Given the eqn x=2, y=sin(t), z=cos(t), draw this function in 3-space.
Homework Equations
ABOVE^
The Attempt at a Solution
I did this:
x^2+y^2+z^2=2^2+(sin(t))^2+(cos(t))^2=5
Therefore we get x^2+y^2+z^2=5
Which is the eqn of a sphere with radius root5.
My friend said it's...
Homework Statement
This is an example problem in my calculus textbook. I don't get how they relate the position vector to the force vector. I have taken a calculus based physics course and I don't remember establishing such relationship between the position and the force vectors.
Note: I have...
Hello, so I have a question that states (these aren't the actual measurements but they are around about the same, I can't remember the exact numbers so I made these up, this way I could apply the same to the actual numbers) an object being 4km above me, 1.4km to the north of me, and 2km to the...
Homework Statement
A projectile is launched at a 60° angle above the horizontal on level ground. The change in its velocity between launch and just before landing is found to be Δv→ = v→landing _ v→launch = -20 y^ m/s .
What is the initial velocity of the Projectile ?
What is its final...
Just wondering why everything measurable in the universe appears in 3 dimensions?(as far as I know!)
And why universe build up(objects) based on three dimensions ?is that anything to do with thermodynamic(Entropy)?
Cheers,
Homework Statement
Determine whether the points lie on straight line
A(2, 4, 2) B(3, 7, -2) C(1, 3, 3)
Homework Equations
The Attempt at a Solution
I've looked up at the equation for lines in three dimension, and it appears to be
x=x_0+at
y=y_0+bt
z=z_0+ct
i tried to take the x y z for A and...