Vector Definition and 1000 Threads

Considering an stopped object in a horizontal plane, the frictional force between the object and the plane would be the product of the friction coefficient (static or kinetic if there was movement between the surfaces) by normal. Since the normal in this case would be given by N (vector) = - mg...

So say I have a bubble embedded in a spacetime with metric: $$ds^2 = -dt^2 + a(t) ( dr^2 + r^2 d\Omega^2_2) $$ how do I compute the normal vector if I assume the wall of the bubble the metric represents follows a time-like trajectory, for any ##a(t)##? Since we are interested in dynamical...

1- Write down the complete MAXWELL equations in differential form and the material equations. 2- An infinitely extensive area is homogeneously filled with a material with a location-dependent permittivity. There are charges in the area. Give the Maxwell equations and material equations of...

I didnt understand the question. The magnitude is 2(pie)/wavelength. I get 78500 rad/cm which is pretty wrong as guess. Where would angle come into picture? Ref: https://www.millersville.edu/physics/experiments/062/index.php Shouldn't direction be like i + j + k ? So will it be like: |k|cos30...

I am having trouble with finding the x and y components of V3 . According to various different websites the correct way to find the components of V3 is Vx=10*cos(100) and Vy=10*sin(100). I can see where the 100 comes from, the previous vector was already traveling 30 degrees and now V3 swung...

Prove, by giving counterexamples, that vector subtraction is not commutative and not associative. ok I read all I could on trying to understand this but seem to not see something simple if we have the example of $u=\begin{bmatrix}2\\-3\\4\\2\end{bmatrix}...

I got the attached photo from someone who solves physics problems on youtube. As you can see their final answer is 6.7i+16j. I understand how she got these values but I came out with something slightly different. I solved for the x and y components on the opposite side of each vector. So...

A = |A|cos33.7 A = |A|0.83195

1.)##\dot{\vec{r}}=\dot{x}\hat{i}+\dot{y}\hat{j}+\dot{z}\hat{k}=\dot{r}\hat{r}## since the unit vector is constant 2.) ##\dot{r}\hat{r}=\frac{x\hat{i}+y\hat{j}+z\hat{k}}{\sqrt{x^2+y^2+z^2}}\frac{\dot{x}x+\dot{y}y+\dot{z}z}{\sqrt{x^2+y^2+z^2}}##...

How do I write the following expression $$\epsilon_{mnk} J_{1n} \partial_i\left[\frac{x_m J_{2i}}{|\vec{x}-\vec{x}'|}\right]$$ back into vectorial form? Einstein summation convention was used here. Context: The above expression was derived from the derivation of torque on a general current...

I am looking at the following document. In section 2.3 they have the formula for the pushforward: f*(X) := Tf o X o f-1 I am having trouble trying to reconcile this with the more familiar equation: f*(X)(g ) = X(g o f) Any help would be appreciated.

Hello (Everyone here has been so helpful -- thank you. Things I thought I knew, I now doubt; and this is so helpful to have this group.) There is an current discussion on Yaw. I am enjoying that. And that raised an issue for me. However, I do NOT want to hijack that thread, so I am posting...

I am considering using a pair of point charges: positive and negative electric charge to model a magnetic dipole's magnetic field by just average the electric field vectors between the two charged particles where they overlap. Will that work? In this case the + field will be vectors pointing...

why the general wave vector q (in the proof of Bloch theorem in Ashcroft Mermin) is represented by k-K, where k is in the 1st BZ ? why not q=k+K ( usual vector form) what is special about k-K?

I have been studying overloading [ ]. I understand how it works from the book. I experiment using vector of structure and obviously I can't make it work. I wonder if it is possible to overload [] on vector of structure. Here is what I have and it's not even close to working, what can I do to...

I am looking at antenna theory and just came upon scalar fields. I found an site giving an example of a scalar field as measuring the temperature in a pan on a stove with a small layer of water. The temperature away from the heat source will be cooler than near it but it doesn't have a...

Summary:: I am suppose to show that this columns matrix does not transform as a vector. In another words, it is not in fact a vector. I think this become trivial if we get the rotation matrix composed of Euler angles. But, i think that it is not the best way to solve this problem, and i...

Solution 1. Based on my analysis, elements of ##V## is a map from the set of numbers ##\{1, 2, ..., n\}## to some say, real number (assuming ##F = \mathbb{R}##), so that an example element of ##F## is ##x(1)##. An example element of the vector space ##F^n## is ##(x_1, x_2, ..., x_n)##. From...

I don't know what is the answer, so i am not sure when to stop the computation or not. The far i reached was ## <k|j> \sum_{j} c_{j}|k>##. That is, the action of the projector operator is, obviously, project the state in |k>. Now, the coefficients was changed. So now what i have to do?

I'm having difficulties solving this. For finding a unit vector that is orthogonal to two unit vectors I understand we use the cross product and such. However, I am confused about how to approach this problem as it has a third vector. We can let x = (x1, x2, x3, x4) be a vector orthogonal to u...

$$\nabla \times B(r)=\frac{\mu _0}{4\pi} \int \nabla \times J(r') \times \frac{ (r-r')}{|r-r|^3}dV'$$ using the vector identity: $$\nabla \times (A \times B) = (B \cdot \nabla)A - B(\nabla \cdot A) - (A \cdot \nabla )B + A(\nabla \cdot B)$$ ##A=J## and ##B=\frac{r-r'}{|r-r'|^3}## since...

Hi, In $\mathbb{R^3} || v-w ||^2=||v||^2 + ||w||^2 - 2||v||\cdot ||w||\cos{\theta}$ But can we say $||v+w||^2=||v||^2 +||w||^2 + 2||v|| \cdot||w|| \cos{\theta}$ where v and w are any two vectors in $\mathbb{R}^3$

That may sound really silly, and that may be due to my lack of understanding of the operations itself, but: if ##|\vec{a}\times\vec{b}|=|\vec{a}|\cdot|\vec{b}|sin\theta##, being ##\theta## the angle between the two vectors, how could ##\vec{b}\times\vec{a}## be different? Wouldn't it be just the...

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\\...

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...