Vectors Definition and 1000 Threads

I understand that a vector space is a set of objects closed under addition and scalar multiplication and satisfies several properties. A functional is a map that takes a vector and produces a scalar. A functional is also called a dual vector. A covector is an object which transforms via the...

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...

I think I've made a mistake with the vectors. I was trying to get to the same point as the original working (made by IWTSE, not me), but one of the sides of a triangle can't be 0. I would appreciate some help.

I want it to be able to add vectors of different length with a 'shift' of c, and where elements do not align to assume a zero in the place of that, My function is: %c is the parameter shift function[sum]=vecadd(y,b,c) p=length(input) q=length(a) if p > q for i= 1+delay:p+delay...

Hi Guys I've attempted the question but not sure if the approach I used is correct. Would someone please have a look at my solution and let me know if it makes sense. Thanks!

I have something I cannot fix in this program. Also I don't know how to test EOF in vector to jump out of do-while loop. //Vector length test #include<iostream> #include <vector> #include <cstring> #include <iomanip> using namespace std; const int ln = 21, al = 31, eA = 21, pN = 15...

In ##\mathbb{R}^2##, there are two lines passing through the origin that are perpendicular to each other. The orientation of one of the lines with respect to ##x##-axis is ##\psi \in [0, \pi]##, where ##\psi## is uniformly distributed in ##[0, \pi]##. Also, there are two vectors in...

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.

Thank you to all those who helped me solve my last question. This week, I've been assigned an interesting problem about toruses. I think I've solved most of this problem on my own, but I'd like to hear a few suggestions for part c. I think this map multiplies tangent vectors by a factor of...

i have attached my attempt above i have sent it to my teacher and he said i should fix the mistakes and resend it

In a certain anisotropic conductive material, the relationship between the current density ##\vec j## and the electric field ##\vec E## is given by: ##\vec j = \sigma_0\vec E + \sigma_1\vec n(\vec n\cdot\vec E)## where ##\vec n## is a constant unit vector. i) Calculate the angle between the...

A coordinate system with the coordinates s and t in R^2 is defined by the coordinate transformations: s = y/y_0 and t=y/y_0 - tan(x/x_0) , where x_0 and y_0 are constants. a) Determine the area that includes the point (x, y) = (0, 0) where the coordinate system is well defined. Express the...

a) Since ##tan(x/x_0)## is not defined for ##x=\pm\pi/2\cdot x_0## I assume x must be in between those values therefore ##-\pi/2\cdot x_0 < x < \pi/2\cdot x_0## and y can be any real number. Is this the correct answer on a)? b) I can solve x and y for s and t which gives me ##y=y_0\cdot s## and...

To my understanding, to get the basis vectors for a given coordinate system (in this case being the elliptic cylindrical coordinate system), I need to do something like shown below, right? $$\hat{\mu}_x = \hat{\mu} \cdot \hat{x}$$ $$\hat{v}_z = \hat{v} \cdot \hat{z}$$ And do that for...

How I would have guessed you were supposed to solve it: What you are supposed to do is just take the gradients of all the u:s and divide by the absolute value of the gradient? But what formula is that why is the way I did not the correct way to do it? Thanks in advance!

I'm reading a section on the derivative of a vector in a manifold. Quoting (the notation ##A^{\alpha}_{\beta'}## means ##\partial x^{\alpha}/\partial x^{\beta'}## - instead of using primed and unprimed variables, we use primed/unprimed indices to distinguish different bases): Now this "we know...

Let's say you have a material element with normal and shear stress. These stresses were computed using stress transformation. When the material deforms, should the normal stress vectors remain normal to the surface (sketch 1) or parallel to the other surface (sketch 2)? Which would be more...

When i read this in the book "A VECTOR APPROACH TO OSCILLATIONS" i was a little shocked, because first it make quotients of vectors, and after this he defines this planar product, i searched this in google: i found nothing. Anyway, this operations make sense if we imagine the vectors...

This is the exact definition and I've summarized it, as I understand it above. Why is it, that for elements in the third subspace, closure will be lost? Wouldn't you still get another vector (when you add two vectors in that subspace), that's still a linear combination of the vectors in the...

On the way to Kruskal coordinates, Carroll introduces coordinates ##\left(v^\prime,u^\prime,\theta,\phi\right)## with metric equation$$ {ds}^2=-\frac{2{R_s}^3}{r}e^{-r / R_s}\left(dv^\prime du^\prime+du^\prime dv^\prime\right)+r^2{d\Omega}^2 $$ ##R_s=2GM## and we're using a ##-+++## signature...

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

Homework Statement:: Wondering about the relationship between gradient vectors, level surfaces and tangent planes Relevant Equations:: . I know that the gradient vector is orthogonal to the level surface at some point p, but is the gradient vector also orthogonal to the tangent vector at that...

Consider the transformation of the components of a vector ##\vec{v}## from an orthonormal coordinate system with a basis ##\{\vec{e}_1, \vec{e}_2, \vec{e}_3 \}## to another with a basis ##\{\vec{e}'_1, \vec{e}'_2, \vec{e}'_3 \}## The transformation equation for the components of ##\vec{v}##...

I'm studying 'Core Principles of Special and General Relativity' by Luscombe - the chapter on tensors. Quoting: The book goes on to talk about a switch to the spherical coordinate system, in which ##\mathbf{r}## is specified as: $$\mathbf{r}=r\sin\theta\cos\phi\ \mathbf{\hat...

I don't really know where to start. Trying to use cosine rule but failed because no information about angle. Thanks

I solved it using parallelogram law if vector addition but didn't got the correct answer.why? Is their any other way to add velocity vectors. How to do this problem

I know that $\arccos{(\cos{\phi_1}\cos{\phi_2}+\sin{\phi_1}\sin{\phi_2}\cos{(\theta_2-\theta_1)})}=\gamma$ But how can i answer the above question? If any member knows the proof of this formula may reply to this question with correct proof.

Hey! 😊 Let $\mathbb{K}$ a field and let $V$ a $\mathbb{K}$-vector space. Let $1\leq m, n\in \mathbb{N}$ and $n=\dim_{\mathbb{K}}V$. Let $v_1, \ldots , v_m\in V$ be linearly independent. Let $\lambda_1, \ldots , \lambda_m, \mu_1, \ldots , \mu_m\in \mathbb{K}$ such that...

Hello all, I was just introduced the Levi-Civita symbol and its utility in vector operations. The textbook I am following claims that, for basis vectors e_1, e_2, e_3 in an orthonormal coordinate system, the symbol can be used to represent the cross product as follows: e_i \times e_j =...

Good day all. Given that in Sean Carroll`s Lectures on GR he states that when calculating the covariant derivative of a 1-Form the Christoffel symbols have a negative sign as opposed to for the covariant derivative of a vector, would it be naive to think that, given the usual equation for the...

Let's consider this scenario. I have two conceptually different video datasets, for example a dataset A composed of videos about cats and a dataset B composed of videos about houses. Now, I'm able to extract a feature vectors from both the samples of the datasets A and B, and I know that, each...

Hey! Let $1\leq n\in \mathbb{N}$, $V=\mathbb{R}^n$ and $\cdot$ the standard scalar multiplication. Let $b_1, \ldots , b_k\in V$ such that $$b_i\cdot b_j=\delta_{ij}$$ Let $\lambda_1, \ldots , \lambda_k\in \mathbb{R}$. Determine $\displaystyle{\left (\sum_{i=1}^k\lambda_i b_i\right )\cdot...

I know that taking the scalar product of the harmonic (Laplacian) friction term with ##\underline u## is $$\underline u \cdot [\nabla \cdot(A\nabla \underline u)] = \nabla \cdot (\underline u A \nabla \underline u) - A (\nabla \underline u )^2 $$ where ##\underline u = (u,v)## and ##A## is a...

Hey! :o Let $1\leq k,m,n\in \mathbb{N}$, $V:=\mathbb{R}^n$ and $U$ a subspace of $V$ with $\dim_{\mathbb{R}}U=m$. Let $u_1, \ldots , u_k\in U$ be linear independent. Show that there are vectors $u_{k+1}, \ldots , u_m\in U$ such that $(u_1, \ldots , u_m)$ is a basis of $U$. Hint: Use the...

Below is the attempted solution of a tutor. However, I do question his solution method. Therefore, I would sincerely appreciate it if anyone could tell me what is going on with the below solution. First off, the rotation of the matrix could be expressed as below: $$G = \begin{pmatrix} AB & -||A...

if someone can concur that'd be great; also, is there any way for me to check myself in the future?

For fun, I decided to prove that two timelike never can be orthogonal. And for this, I used the Cauchy Inequality for that. Such that The timelike vectors defined as, $$g(\vec{v_1}, \vec{v_1}) = \vec{v_1} \cdot \vec{v_1} <0$$ $$g(\vec{v_2}, \vec{v_2}) = \vec{v_2} \cdot \vec{v_2} <0$$ And the...

Hi I am a beginner in this topic. I didn't understand this question type clearly.What does it mean" With Magnitude and Unit Vectors" exactly? May you help me for the solution step by step :). Thanks in advance.

Let we have ##J_i \in{J_1,J_2,J_3}## ,and ##K_i \in{K_1,K_2,K_3}##, rotation and boost generators respectable . ##A_i=\cfrac{1}{2}(J_i+iK_i)##, and ##[A_i,A_j]=i\epsilon_{ijk}A_k## ##[K_i,K_j]=-i\epsilon_{ijk}J_k## ##[J_i,K_j]=-i\epsilon_{ijk}K_k## How proof that ##(m,n)A_i=J^{(m)}_i\otimes...

The proof that the set is a subspace is easy. What I don't get about this exercise is the dimension of the subspace. Why is the dimension of the subspace ##n-1##? I really don't have a clue on how to go through this.

I know that the mathematical form of the line element of spacetime is invariant in all inertial reference frames, namely $$ds^2 = -(cdt^2) + dx^2 + dy^2 + dz^2$$ From what I understand, the actual spacetime distance between two events is the same numerical quantity in all reference frames...

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But the answer in my book is given that sec(theta) =3. Where am I going wrong?

I'm looking forward to have a better understanding of the polarization vector in quantum field theory in order to solve a particular problem. In class and in several textbooks I see that ##s^\mu=(0,\vec s)## and ##|\vec s|=1##. Are polarizations vectors defined to have no temporal component in...