What is Vector: Definition and 1000 Discussions

The Poynting vector $$\vec S=\frac{1}{\mu_0} \vec E \times \vec B$$ gives the power per unit area. If I need this in terms of electric field only,I should be able to write B=E/c (for EM wave) Assuming they're perpendicular, ##S =\frac{1}{\mu_0 c}E^2##. Now, ##c=\frac{1}{\sqrt{\mu_0 \epsilon_0}}...

hi guys this seems like a simple problem but i am stuck reaching the final form as requested , the question is given the magnetic vector potential $$\vec{A} = \frac{\hat{\rho}}{\rho}\beta e^{[-kz+\frac{i\omega}{c}(nz-ct)]}$$ prove that $$B = (n/c + ik/\omega)(\hat{z}×\vec{E})$$ simple enough i...

Be ##T_{1}, T_{2}## upper and lower matrix, respectivelly. Show that we haven't matrix ##M(NxN)## such that ##M(NxN) = T_{1}\bigoplus T_{2}## I am not sure if i get what the statement is talking about, can't we call ##T_{1},T_{2} = 0##? Where 0 is the matrix (NxN) with zeros on all its entries...

Summary:: Problem interpreting a vector space of functions f such that f: S={1} -> R Hello, Another question related to Jim Hefferon' Linear Algebra free book. Before explaining what I don't understand, here is the problem : I have trouble understanding how the dimension of resulting space...

Summary:: Be the set X of vectors {x1,...,xn} belong to the vector space E. If this set X is convex, prove that all the convex combination of X yet belong to X. Where convex combination are the expression t1*x1 + t2*x2 + ... + tn*xn where t1,...,tn >= 0 and t1 + ... + tn = 1 I tried to suppose...

How can I prove that Zp is a vector space if and only if p is prime

Hello, I am doing a vector space exercise involving functions using the free linear algebra book from Jim Hefferon (available for free at http://joshua.smcvt.edu/linearalgebra/book.pdf) and I have trouble with the author's solution for problem II.1.24 (a) of page 117, which goes like this ...

Apologies in advance if I mess up the LaTeX. If that happens I'll be editing it right away. By starting off with ##\nabla^{'}_{\mu} V^{'\nu}## and applying multiple transformation laws, I arrive at the following expression $$ \frac{\partial x^{\lambda}}{\partial x'^{\mu}} \frac{\partial...

I am now learning C ++ and trying to learn class and vector. I'm trying to write code, but I got an error. this is my class and enum class: enum class state: char{ empty='.', filled_with_x='x', filled_with_o='o'}; class class1{ private: class class2{ class2()...

The answer is D (60 degrees) and I understand how to get that answer. But this assumes that the new velocity's component of v/4 can form right angles with another component of the new velocity. So I'm confused whether vector components always form right angles to each other. When I searched...

Writing both ##\vec{U}## and ##\vec{B}## with magnitude in all the three spatial coordinates: $$ \vec{U}\times \vec{B}= (U_{x}\cdot \widehat{i}+U_{y}\cdot \widehat{j}+U_{z}\cdot \widehat{k})\times (B_{x}\cdot \widehat{i}+B_{y}\cdot \widehat{j}+B_{z}\cdot \widehat{k})$$ From this point on, I...

The direction of the magnetic potential, ##\vec A##, must be in the direction of the current, which is in ##\hat z## direction in cylindrical coordinates. It is obvious that the potential only varies with ##s##. Therefore, $$\vec A = A(s) \hat z$$ Therefore, $$\nabla \times \vec A = \vec B$$...

The question I am trying to solve is what is the velocity vector (direction and magnitude) of an object in 2 d space. We know the distance measured to the car from two different angles. We know the radial velocity of the car on both measurements. The radial velocity is the component of the...

This week, I've been assigned a problem about a 3-sphere. I am confused how to approach this problem and any comments would be greatly appreciated. (a) - would I be correct to assume the metric G is simply the dot product of two vector fields with dx^2 dy^2 du^2 and dv^2 next to their...

$\tiny{1.5.12}$ Describe all solutions of $Ax=0$ in parametric vector form, where $A$ is row equivalent to the given matrix. RREF $A=\left[\begin{array}{rrrrrr} 1&5&2&-6&9& 0\\ 0&0&1&-7&4&-8\\ 0& 0& 0& 0& 0&1\\ 0& 0& 0& 0& 0&0 \end{array}\right] \sim \left[\begin{array}{rrrrrr} 1&5&0&8&1&0\\...

Hello all, I have produced a contour plot of a given function f. To this, I have added the vector field (arrows) for the gradient of f calculated analytically. I have then also added the vector field numerically by using the np.gradient function in python. In both cases, the vector field...

Describe all solutions of $Ax=b$ in parametric vector form, where $A$ is row equivalent to the given matrix. $A=\left[\begin{array}{rrrrr} 1&-3&-8&5\\ 0&1&2&-4 \end{array}\right]$ RREF $\begin{bmatrix}1&0&-2&-7\\ 0&1&2&-4\end{bmatrix}$ general equation $\begin{array}{rrrrr} x_1& &-2x_3&-7x_4...

If say we have a scalar function ##T(x,y,z)## (say the temperature in a room). then the rate at which T changes in a particular direction is given by the above equation) say You move in the ##Y##direction then ##T## does not change in the ##x## and ##z## directions hence ##dT = \frac{\partial...

Write the solution set of the given homogeneous systems in parametric vector form. $\begin{array}{rrrr} -2x_1& +2x_2& +4x_3& =0\\ -4x_1& -4x_2& -8x_3& =0\\ &-3x_2& -3x_3& =0 \end{array}\implies \left[\begin{array}{rrrr} x_1\\x_2\\x_3 \end{array}\right] =\left[\begin{array}{rrrr}...

In part c, plotting the vector field shows the vector field is symmetric in x and y in the sets {x=y}. in {x=y}, the variables can be interchanged and the solution becomes x = x°e^t y = y°e^tHowever, these solutions do not work for anywhere except {x=y} and don't satisfy dx/dt = y and dy/dt =...

Doing a review for my SAT Physics test and I'm practicing vectors. However, I am lost on this problem I know I need to use trigonometry to get the lengths then use c^2=a^2+b^2. But I need help going about this.

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Suppose I have a three dimensional unit Vector A and two other unit vectors B and C. If B is rotated a certain amount in three dimensions to get vector C, how do I find what the new Vector D would be if I rotated Vector A the same direction by same amount?

It seems most people say that a vector is either contravariant or covariant. To me it seems like contra/covariance is a property of the components of a vector (with respect to some basis) and not of the vector itself. Any basis {bi} has a reciprocal basis {bi} and any vector can be expressed...

##4\pi\mathcal L = -\mathcal e \frac{\partial A_i }{\partial t} - \phi\mathcal E^i{}_{,i} -\frac{1}{2}N\gamma^{\frac{1}{2}}g_{ij}(\mathcal E^i \mathcal E^j +\mathcal B^i\mathcal B^j) +N^i [ijk]\mathcal E^i\mathcal B^j## MTW (21.100) I'm trying to produce the result required by the problem...

Hi, I apologise as I know I have made similar posts to this in the past and I thought I finally understood it. However, this solution seems to disagree on a technicality. I know the answer ends up as 0, but I still want to understand this from a conceptual point. Question: Evaluate the line...

Hi I made an attempt at this problem but have got the wrong answer The correct answer is actually resultant force = 21.767 N at 61.34 degrees (or 151.34 degrees bearing), but I don't know how they got that? Any help would be appreciated! Thanks

Untill now i have only been able to derive the equations of motion for this lagrangian when the field $$\phi$$ in the Euler-Lagrange equation is the covariant field $$A_{\nu}$$, which came out to be : $$-M^2A^{\nu} = \partial^{\mu}\partial_{\mu}A^{\nu}$$ I have seen examples based on the...

If I'm given a set of four vectors, such as A={(0,1,4,2),(1,0,0,1)...} and am given another set B, whose vectors are given as a form such as (x, y, z, x+y-z) all in ℝ, what steps are needed to show A is a basis of B? I have calculated another basis of B, and found I can use linear combinations...

I have these equations in my book, but I don't know how I can use them in this problem Electric field of a plane has surface electric density σ: E = σ/2εε₀ Ostrogradski - Gauss theorem: Φ₀ = integral DdS Can someone help me :((

I've looked it up online and someone did t=40−65=0.15(h) I was just wondering why they would subtract the velocities. Could something explain this to me please? thanks.

Broke it into its components finding d1x, d1y, d2x, etc... Using those components I found drx to be 228.38km and dry to be 120.429km. Did Pythagoras to get 258km as the resultant displacement, heading N62W. I'm honestly lost. I'm doing the question the correct way, I just don't know what I'm...

I am not completely sure what the formulas ##v_j = v^a\frac {\partial x^j} {\partial \chi^a}## and ##v^b = v^a\frac {\partial \chi^b} {\partial x^j}## mean. Is ##v_j## the j:th cartesian component of the vector ##\vec v## or could it hold for other bases as well? What does the second equation...

Well, I understand that according to the conservation of momentum the total momentum of a system is conserved for objects in an isolated system, that is the sum of total momenta before the collison is equal to the sum of momenta after the collision. In this case, the momentum of the object...

hi i was recently introduced to the Dirac notation and i guess i am following it really well , but can't get my head around the idea that the bra vector said to live in the dual space of the ket vectors , i know about linear transformation and the structure of the vector spaces , and i realize...

I was content with the understanding of the Fourier transform (FT) as a change of basis, from the time to the frequency basis or vice versa, an approach that I have often seen reflected in texts. It makes sense, since it is the usual trick so often done in Physics: you have a problem that is...

Here is how my teacher solved this: I understand what the nabla operator does, ##∇\cdot\vec v## means that I am supposed to calculate ##\sum_{n=1}^3\frac {d\vec v} {dx_n}## where ##x_n## are cylindrical coordinates and ##\vec e_3 = \vec e_z##. I understand why ##∇\cdot\vec v = 0##, I would get...

##(\nabla\times\vec B) \times \vec B=\nabla \cdot (\vec B\vec B -\frac 1 2B^2\mathcal I)-(\nabla \cdot \vec B)\vec B## ##\mathcal I## is the unit tensor

I'm reading 'Core Principles of Special and General Relativity' by Luscombe - the part on parallel transport. I guess ##U^{\beta}## and ##v## are vector fields instead of vectors as claimed in the quote. Till here I can understand, but then it's written: I want to clarify my understanding of...

I've taken multivariable/vector calc and can do most of the basic operations and have an OK understanding of the fundamental concepts, but certainly can't "see it" like I can calc I and II. In those subjects, I often feel competent to take on any problem I come across because the concepts are...

For divergence: We learned to draw a circle at different locations and to see if gas is expanding/contracting. Whenever the y-coordinate is positive, the gas seems to be expanding, and it's contracting when negative. I find it hard to tell if the gas is expanding or contracting as I go to the...

I have not tried to make any calculation. It's nonsense, because I don't understand the statement. The first vector points to the west. Given a two dimensional coordinate system, the first vector is pointing to the left. I imagine geographical coordinates, north (+y), south (-y), west (-x), and...

I'm reading 'Core Principles of Special and General Relativity' by Luscombe, specifically the introductory section on problems with defining usual notion of differentiation for tensor fields. I'll quote the relevant part: Since the equation above is a notational mess, here's my attempt to...

Seems to me the answer is a specific vector: The second forms a plane, while the first X is just a vector. The intersection between the λX that generates the (properties of all vectors that lie in the...) plane (i am not saying X is the director vector!) How to write this in vector language?

I was just thinking, if is said to me demonstrate any geometry statement, can i open the vector in its vector's coordinates? I will say more about: For example, if is said to me: Proof the square's diagonals are orthogonal, how plausible is a proof like?: d1 = Diagonal one = (a,b,c) d2 =...

I managed to expand a general expression from the alternatives that would leave me to the answer, that is: I will receive the alternatives like above, so i find the equation: C = -sina, P = cosa So reducing B: R: Reducing D: R: Is this right?

is it true that vectors are symbolised as an italic boldface 'a'

Hi PF! I am trying to multiply each component of B by the matrix A and then solve A\C. See the code below. A = rand(4); B = rand(5,1); C = rand(4,1); for i = 1:5 sol(:,i) = (B(i)*A)\C end But there has to be a way to do this without a for-loop, right? I'd really appreciate any help you have!

For my understanding, to move to the coolest place, it has to move in direction of -∇f(x,y) How can I find the value of 'k' to evaluate the directional derivative and what can I do with the vertices given.