What is Vector: Definition and 1000 Discussions

Hey! :o We consider the $\mathbb{F}_2$-vector space $(2^M, +, \cap)$, where $M$ is non-empty set and $+ : 2^M\times 2^M \rightarrow 2^M: (X,Y)\mapsto (X\cup Y)\setminus (X\cap Y)$. I want to show that $(2^M, +, \cap )$ for $\mathbb{K}=\{\emptyset , M\}$ satisfies the axioms of a vector space...

Hey! :o Let $1\leq n\in \mathbb{N}$ and let $U_1, U_2$ be subspaces of the $\mathbb{R}$-vector space $\mathbb{R}^n$. I want to prove or disprove the following: The set $\{f\in \mathbb{R}^{\mathbb{R}} \mid \exists x\in \mathbb{R} : f(x)=0_{\mathbb{R}}\}$ is a subspace of...

I'm working through Lahiri & Pal's book A First Book of Quantum Field Theory, Second Edition and I'm stuck on their explanation of the polarization vector in quantum electrodynamics in Chapters 8 and 9. In section 8.8, they derive a formula for the sum over the transverse polarization modes of...

Is there some rule or standard that determines whether we define a vector with upper or lower case? I have not been told of any particular rule but it seems with velocity and acceleration they are lower case but force has always been upper case from what I've been reading so far. Is there a...

If I have a vector field say ## v = e^{z}(y\hat{i}+x\hat{j}) ##, and I want to calculate the divergence. Do I only take partial derivatives with respect to x and y (like so, ## \frac{\partial A_x}{\partial x} + \frac{\partial A_y}{\partial y} ##) or should I take partial derivatives with respect...

Homework Statement Let D be the triangle with vertices (0,0), (1,0) and (0,1). Evaluate: ∫∫exp((y-x)/(y+x))dxdy for D by making the substitutions u=y-x and v=y+x Homework EquationsThe Attempt at a Solution So first I found an equation for y and x respectively: y=(u+v)/2 and x=(v-u)/2 Then...

The question goes like: find the SA of the portion S of the cone z^2 =x^2 +y^2 where z>=0 contained within the cylinder y^2+z^2<=49 this is my attempt using the formula for SA, I could switch to parametric eqns, but even then I'd have hard time setting up limits of integration.

Hi guys, Consider a circular capacitor with a disk of radius a and plate separation d, as shown in the figure below. Assuming the capacitor is filled with a dielectric constant epsilon and the capacitor is fed by a time harmonic current I0 (a) Find the magnetic field distribution inside the...

Let's say i have 2 arbitrary vectors in a 3d space. I want to project Vector A to Vector B using a specified normal. edit: better image A is green, B is red, C is red arrow. Blue is result. In this case, i want to project green vector to red vector in the red direction. This would give me...

Homework Statement A proton moves with a speed ##v = 3 \cdot 10^5 \frac{m}{s}## in the parallel direction to ##i+k##. A magnetic field of ##1T##, in the ##i+j+k## acts over it. Which electric field must we apply in this region so that the Lorentz force over the proton is null? Homework...

Homework Statement Solve the following vector equation for ##\vec{y}##. ##\vec{a}##, and ##\vec{b}## are linearly independent vectors of the three dimensional space. ##\vec{a} \times (8\vec{y}+\vec{b}) = \vec{b}\times(-5\vec{y}+\vec{a})## Homework EquationsThe Attempt at a Solution First I...

Homework Statement Consider the real-vector space of polynomials (i.e. real coefficients) ##f(x)## of at most degree ##3##, let's call that space ##X##. And consider the real-vector space of polynomials (i.e. real coefficients) of at most degree ##2##, call that ##Y##. And consider the linear...

Is it possible to introduce the concept of a gradient vector on a manifold without a metric?

Homework Statement In this problem, we'll construct the ##(\frac{1}{2},\frac{1}{2})## representation which acts on "bi-spinors" ##V_{\alpha\dot{\alpha}}## with ##\alpha=1,2## and ##\dot{\alpha}=1,2##. It is convential, and convenient, to define these bi-spinors so that the first index...

Is it possible to expand a state vector in a basis where the basis vectors are not eigenvectors for some observable A? Or must it always be the case that when we expand our state vector in some basis, it will always be with respect to some observable A?

I'm learning APL and this is how a vector is defined https://tryapl.org: All data resides in arrays. An array is a rectangular collection of numbers, characters and arrays, arranged along zero or more axes. We can use more specific terms for some arrays, like a single number is a scalar, a list...

For example, does this always hold true? M_ab = v_a × w_b If not, where does it break down?

Suppose a plane contains the origin and has normal $n$. Is it true that the projection of a vector $u$ on the plane along vector $v$ is $(v\times u)\times n$, where $\times$ denotes the cross product? I can see that the direction is right, but I am not sure about the length. Links to textbooks...

Homework Statement a. Find a point at where these lines intersect b. Find the equation of a plane that contains the two lines. Homework Equations r[/B] = <1,3,0> + t<3,-3,2> r = <4,0,2> + s<-3,3,0> The Attempt at a Solution I correctly found the point of intersection to be...

If the radius unit vector is giving us some direction in spherical coordinates, why do we need the angle vectors or vice versa?

Given some metric, what is an example where the length of a vector is not preserved?

Homework Statement Convert the vector given in spherical coordinates to cylindrical coordinates: \vec{F}(r,\theta,\varphi) = \frac{F_{0}}{arsin\theta}\bigg{[}(a^2 + arsin\theta cos\varphi)(sin\theta \hat{r} + cos\theta \hat{\theta}) - (a^2 + arsin\theta sin\varphi - r^2 sin^2\theta)...

I'm having a little trouble with this : We have ##(\alpha\vec{a})\cdot b = \alpha(\vec{a}\cdot\vec{b})## but shouldn't it be ##|\alpha|(\vec{a}\cdot\vec{b})## instead since ##||\alpha \vec{a}||=|\alpha|.||\vec{a}||## ? ##(\alpha\vec{a})\cdot b = ||\alpha\vec{a}||.||\vec{b}||.\cos\theta =...

Hello I have a question if it possible, Let X a tangantial vector field of a riemannian manifolds M, and f a smooth function define on M. Is it true that X(exp-f)=-exp(-f).X(f) And div( exp(-f).X)=exp(-f)〈gradf, X〉+exp(-f)div(X)? Thank you

Homework Statement A Mercedes-Benz 300SL (m = 1700 kg) is parked on a road that rises 15 degrees above the horizontal. What are the magnitudes of (a) the normal force and (b) the static frictional force that the ground exerts on the tires? Important: Assume that the road is higher up to the...

Hi. Is it true to say that the integral over all volume of ∇ψ where ψ is a scalar function of position and time is just ψ ? Thanks

Homework Statement Give an example of the associative property of vector addition using vectors in Cartesion form. Homework Equations (u+v)+w=u+(v+w) The Attempt at a Solution I can't figure out how to get the arrow on top of my work so I wrote it without it. I'm somewhat confused on why I...

the question: My attempt: The partial derivatives did not match so i simply tried to find f(x,y) I got the set of equations on the right but that's about it.

Homework Statement For any vector in 2D space, it can be broken down into its horizontal and vertical components. Homework Equations In one of my engineering classes, we are using the following equation to determine the magnitude of a vector: $$u=v_1 \cdot cos\theta +u_2 \cdot sin\theta$$...

1 Is Equilibrium a Scalar or Vector quantity? 2 What is the unit of Equilibrium? Thanks & Regards, Prashant S Akerkar

Homework Statement Forces of 11.8N north, 19.2N east, and 15.9N south are simultaneously applied to a 3.93kg mass as it rests on an air table. What is the magnitude of its acceleration? What is the direction of the acceleration in degrees? (Take east to be 0 degrees and counterclockwise to be...

nmh{796} $\textsf{Suppose $Y_1$ and $Y_2$ form a basis for a 2-dimensional vector space $V$ .}\\$ $\textsf{Show that the vectors $Y_1+Y_2$ and $Y_1−Y_2$ are also a basis for $V$.}$ $$Y_1=\begin{bmatrix}a\\b\end{bmatrix} \textit{ and }Y_2=\begin{bmatrix}c\\d\end{bmatrix}$$ $\textit{ then }$...

What's the difference between magnitude and size? I mean, how can I say that AB vector has a magnitude of 9 Newton, and its length is 4 cm.

Homework Statement Take ∂2E/∂t2 E(r,t)=E0cos((k(u^·r−ct)+φ) in which u^ is a unit vector. Homework Equations d/dx(cosx)=-sinx The Attempt at a Solution I had calc 3 four years ago and can't for the life of me remember how to differentiate the unit vector. I came up with...

what is the use of multiplying and dividing a vector by a scalar?

Homework Statement "A bird flew 40 m to the west, then 100 m in a direction 36.9 degrees to the north of east. Use the algebraic addition of vectors to fins the magnitude of the bird's net displacement" Use sin (36.9) =0.6 and cos (36.9) = 0.8 Homework Equations Vector addition? A =...

Why are the resultant's X and Y components of two forces equal the sum of the X and Y components of the two forces?

Homework Statement Three vectors are given: A=2i+3j, B=1i+5j, C=-1i+3j Find constants x and y such that xA+yB=C Homework Equations N/A The Attempt at a Solution The form of the final equation reminded me if standard form of a slope, so I found the total vector for A,B, and C. I was then going...

When doing integration such as \int_{0}^{2\pi} \hat{\rho} d\phi which would give us 2\pi \hat{\rho} , must we decompose \hat{ρ} into sin(\phi) \hat{i} + cos(\phi) \hat{j} , then \int_{0}^{2\pi} (sin(\phi) \hat{i} + cos(\phi)\hat{j}) d\phi , which would give us 0 instead? Thanks

Homework Statement Givens: An object with a mass of 2kg has a momentum of p=<-1,-2,3>. The first two questions asked for the magnitude of the momentum and the corresponding unit vector, which i found to be 3.74 and <-0.267,-0.535,0.802> respectively. The next question asks for the speed of the...

Homework Statement Vectors A and B have equal magnitudes of 4.93. If the sum of A and B is the vector 6.79j, determine the angle between A and B Homework Equations c^2 = a^2 + b^2 -2abcos(theta) The Attempt at a Solution I just rearranged the formula above so that I could solve for the...

I have this odd difficulty when it comes to vector calculus. No matter what I do, seeing these equations just stays daunting. Its a massive effort for me to read them, and when I say "read" I mean look at the equations, and actually understand what it is it wants me to do. Now as with most math...

Homework Statement Two vectors A and B have precisely equal magnitudes. For the magnitude of A + B to be 65 times greater than the magnitude of A - B, what must be the angle between them? Homework EquationsThe Attempt at a Solution I tried using the dot product and solving for the angle but i...

Homework Statement r1= 2 7 r2=-1 -6 Homework Equations A-lambda*I=0 (A-lambda*I)*x=0 The Attempt at a Solution I have got following eigen values: lambda1 = -5 and lambda2=1 A-lambdaI matrix is: r1 = 7 7 r2 = -1 -1 and x matrix is: r1 =x r2 =y I can't understand why we have to use...

What's the difference between saying that a vector's direction is north of east and north east?

I just want to know the difference between those rules: 1. R^2 = F1^2 * F2^2 + 2*F1*F2*COS(the angle between F1 and F2) 2. The second is about the parallelogram rule, it says that the two vectors are added and their summation is the magnitude of the resultant. Which one is correct?

Hello, We defined a k-form on a smooth manifold M as a transfromation Where the right space is the one of the alternating k-linear forms over the tangent space in p. If we suppose we know, that we get a basis of this space by using the wedge-product and a basis of the dual space, then we might...

I have a question regarding the dot product and the cross product differentiation. I was wondering whether: $$\frac{d(\vec{A}.\vec{B})}{du} = \vec{A}. \frac{d\vec{B}}{du} + \frac{d\vec{A}}{du} .\vec{B}$$ is the same as $$\frac{d(\vec{A}.\vec{B})}{du} = \frac{d\vec{A}}{du} .\vec{B} + \vec{A}...

Homework Statement Consider a configuration consisting one +q charge ( upper right) and three −q charges, arranged in a square. Side lengths = d. Calculate the total F force vector acting on charge +q.Homework Equations Vector form of culomb’s force F=( kq1q2/r^2) rhat (rhat for unit...

Homework Statement Problem attached in Dipole.JpgHomework Equations The Attempt at a Solution I am fine with Part A of the problem. I am just trying to understand what part B is asking. Are they asking what is the magnitude and direction of the dipole vector associated with charges q1 and q1 at...