What is Vector: Definition and 1000 Discussions

Homework Statement Recall that F is the vector space of functions from ℝ to ℝ, with the usual operations of addition and scalar multiplication of functions. For each of the following subsets of F, write down two functions that belong to the subset, and determine whether or not the subset is a...

Homework Statement [/B] 1. Suppose {v1, . . . , vk} is a linearly independent set of vectors in Rn and suppose A is an m × n matrix such that Nul A = {0}. (a) Prove that {Av1, . . . , Avk} is linearly independent. (b) Suppose that {v1, . . . , vk} is actually a basis for Rn. Under what...

As far as I can tell, in GR, the Chirstoffel symbol in the expression of the Connection is analogous to the vector potential, A, in the definition of the Covariant Derivative. The Chirstoffel symbol compensates for changes in curvature and helps define what it means for a tensor to remain...

Homework Statement Show that (ap1,bp2) is not a vector unless a = b.(The 1 and 2 are superscripts) In Einstien's Gravity in a Nutshell, p 43, Zee states the above is not a vector because it doesn't transform like a vector under rotation. When I use the usual rotation matrix for rotation about...

In string theory, if we have NN BCs along ##X^i, i = 1, \ldots, n-1## and DD BCs along ##X^a, a = n, \ldots, 25## then you get, from ##\alpha^{i,a}_{-1}|0,p\rangle ##, ##n## massless vectors and ##24-n## massless scalars. I understand that for the first excited level, ##M^2=0## and so we have...

Hello Forum and happy new year, Aside from a rigorous definitions, a linear vector space contains an infinity of elements called vectors that must obey certain rules. Based on the dimension ##N## (finite or infinite) of the vector space, we can always find a set of ##n=N## linearly independent...

see post 3

$\tiny{s6.194.4.12.4.29}$ $\textsf{a. Find a nonzero vector orthogonal to plane through the points: }$ $\textsf{b. Find the area of the triangle PQR}$ \begin{align} \displaystyle &P(1,0,0)& &Q(0,2,0)& &R(0,0,3)\\ %&=\color{red}{\frac{1209}{28} } \end{align} $\textit{do what first?}$

Hello Forum, The state of a quantum system is indicated by##\Psi## in Dirac notation. Every observable (position, momentum, energy, angular momentum, spin, etc.) corresponds to a linear operator that acts on ##\Psi##.Every operator has its own set of eigenstates which form an orthonormal basis...

Finding length of vector with unknown variable. Purely for study purposes. Find the smallest possible length of the vector →v . Let vector V = (-2/3,b,16/7) Equation for finding length of vector : Sqrt(a^2+b^2+c^2)Question would be quite straight forward had there been no unknown variable but...

I am a little bit confused on Center of Mass and Center of Gravity about what are they vectors or scalars . As I may think they are neither because they are simply two points . Am I saying right or wrong .

I have learned that if I multiply a vector, say 3i + 4j, by a scalar that is a real number, say 2, the effect of the operation is to expand the size of the magnitude of the original vector, by 2 in this case, and the result would be 6i + 8j. What would be the effect on a vector, like 3i + 4j...

Given the Fourier conjugates ##\vec{r}## and ##\vec{k}## where ##\vec{r} = [r_1,r_2,r_3]## and ##\vec{k} = [k_1,k_2,k_3]## , are ##r_1## /##k_1##, ##r_2##/##k_2##, ##r_3##/##k_3## also Fourier conjugates, such that: ##\begin{equation} \begin{split} f(\vec{r})&=[f_1(r_1),f_2(r_2),f_3(r_3)] \\...

A car is driven eastward for a distance of 50 miles, then northward for 30 miles, and then in a direction 30 degree east of north for 25 miles. Draw the vector diagram and determine the total displacement of the car from its starting point. The relevant equations I used here are: i) rx= ax +...

Homework Statement Find the value of ez if e = 1/3i + 2/3j + ezk Homework Equations Any help? The Attempt at a Solution Have no idea where to start

Hi everyone please help how can i find the force in Q(a)(ii)?

How do you solve this question I just need a solution

Homework Statement Homework Equations The Attempt at a Solution I am really lost here because our professor gave us no example problems leading up to the final exam and now we are expected to understand everything about vector calculus. This is my attempt at the cross product and...

Homework Statement Given the velocity vector in the polar coordinates, ##\vec v=-awsin{wt}\vec e_r + awcos{wt}\vec e_\theta## determine the average velocity vector and velocity intensity over a time period ##[0, \pi/2w]## Homework Equations 3. The Attempt at a Solution [/B] For the first part...

1. Homework Statement I am looking at problem 2.2 pictured above. I have solved all portions of the question except the last part, which asks for the angle between the normal vector to the surface and the z-axis. I am aware that the normal vector is simply equal to the gradient of the surface...

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 recognize the rate of change of a vector in an inertial frame S can be related to the rate of change of the vector in a rotating frame S0 by the equation below taken from my textbook, where Ω is the angular velocity vector. $$\Big(\frac{dQ}{dt}\Big)_{S_{0}}= \Big(\frac{dQ}{dt}\Big)_{S} +...

I assume this is a simple summation of the normal components of the vector fields at the given points multiplied by dA which in this case would be 1/4. This is not being accepted as the correct answer. Not sure where I am going wrong. My textbook doesn't discuss estimating surface integrals...

Hey! :o I want to check if the following are true. $V_1=\{a\in \mathbb{R}\mid a>0\}$ with the common multiplication as the vector addition and the scalar multiplication $\lambda \odot v=v^{\lambda}$ is a $\mathbb{R}$-vector space. $V_2=\{(x,y)\in \mathbb{Q}^2 \mid x^2=-y^2\}$ with the...

In our section on path independence, we were asked to find the potential function given a vector field. Our teacher says to use only line integrals to find the potential function, and not any other method. Like if we have ##F=\left< M,N,P \right> ## The first step is to determine if the domain...

Homework Statement Bead on spoke: constant speed ##u## along spoke it starts at center at ##t=0## angular position is given by ##\theta=\omega t##, where ##\omega## is a constant Homework Equations ## \frac{d\hat r}{dt} = \dot \theta \hat \theta ## (1) ## \frac{d\hat \theta}{dt} = -\dot...

Sorry may i ask a question here~ i don't understand how did GRIFFITHS prove the statement from(C) to (B) What is his logic in this case? and What is Ai Aii Bi Bii in the integral? thank you

Homework Statement If an object had been projected horizontally with the same magnitude as in the depicted situation, how would the motion compare with that of the object in the diagram? (I have drawn the diagram in my attachment and have done questions c and d but I don't understand question...

When I induce magnetic flux through a closed loop, I should expect the lines of flux produced by current in that loop to oppose the change of flux through that loop. But what happens when that loop, say a rectangular loop, is curved into the shape of the letter J (like a candy cane) and my flux...

http://www.physicspages.com/2014/08/22/electromagnetic-waves-in-vacuum/ Sorry i have no idea how to get the last step in the vector form how to convert and mix it?thank you

Homework Statement Prove that the relationship between the dimensionless amplitude a of a wave and the intensity I of the wave is given by: Homework Equations B = curl (A) B: Magnetic Field A = [mass of electron * c / e ]*a*e^[i*(kx-wt)] where k is the wavenumber and is given by: 2pi /...

Homework Statement Loop of current ##I## sitting in the xy plane. Current goes in counter clockwise direction as seen from positive z axis. Find: a) the magnetic dipole moment b) the approximate magnetic field at points far from the origin c) show that, for points on the z axis, your answer is...

Homework Statement find the vector in R3 that is a unit vector that is normal to the plane with the general equation x − y + √2z=5 [/B]Homework EquationsThe Attempt at a Solution so the orthogonal vector, I just took the coefficients of the general equation, giving (1, -1, √2)[/B] then...

Homework Statement 23. In a ABCD quadrilateral let P,Q,R,S be midpoints of sides AB,BC,CD and DA. Let X be the intersection of BR and DQ, and let Y be the intersection of BS and DP. If ##\vec{BX}=\vec{YD} ## show that ABCD is a parallelogram . Homework Equations ## (\vec{a}\cdot\vec{b})=0##...

<<Mentor note: Moved from non-homework forum>> If a uniform magnetic field ##{\bf{B}}=B_{z}{\bf{\hat{z}}}## exists in a hollow cylinder (with the top and bottom open) with a radius ##R## and axis pointing in the ##z##-direction, then the vector potential...

Homework Statement How does the unit vector have no units I know that the unit vector has a length of 1 and zero units. A representation would be I-hat j-hat k-hat(depending on the coordinate system). But the unit vector is the vector/magnitude. If the unit vector is its vector/magnitude then...

We all know that poynting vector could be used to calculate energy transferred in AC system, but I want to know if that theorem could also be applied to DC current? How to prove this? And how to prove energy transferred in AC system could be calculated via poynting vector?

Let's say we have a vector field that looks similar to this. Assume that the above image is of the x-y plane. The vector arrows circulate a central axis, you can think of them as tangents to circles. The field does not depend on the height z. The lengths of the arrows is a function of their...

Homework Statement I am very rusty on my mathematics and I am wondering if there is a way to express Rs in terms of r1 and r2. The positions of the bodies are all relative to the origin 0 (C.o.M between m1 and m2). Basically I'm trying to express the two vectors coming from m3 in terms of the...

Homework Statement Find the unit tangent vector T(t) for vector valued function r(t) = (e^t)(cos t ) i + (e^t)(sin t ) j + (e^t) k Homework EquationsThe Attempt at a Solution i gt stucked here ... , the ans is [1/ sqrt (3) ] [ (cos t -sin t ) i + (sin t + cos t ) j +k) [/B]

Homework Statement Find Parametric equation and vector equation for the portion of parabola y = 1+(x^2) from (1,2) to (2,5) Homework EquationsThe Attempt at a Solution m = (2-1) i +(5-2)j = i +3j x = t , y = 1+t^2 r(t) = ti + (2=3t)j, where 0<t<1 , but the ans given is r(t) = ti + (1+t^2) j...

Currently working through some exercises introducing myself to quantum field theory, however I'm completely lost with this problem. Let $$L$$ be a Lagrangian for for a real vector field $$A_\mu$$ with field strength $$F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu$$ gauge parameter...

Let ##0##-form ##f =## function ##f## ##1##-form ##\alpha^{1} =## covariant expression for a vector ##\bf{A}## Then consider the following dictionary of symbolic identifications of expressions expressed in the language of differential forms on a manifold and expressions expressed in the...

Homework Statement A particle starting from rest revolves with uniformly increasing speed in a clockwise circle in an xy plane. The center of the circle is at the origin of an xy coordinate system. At t=0, the particle is at x=0.0m, y=2.0m. At t=2.0s, it has made one-quarter of a revolution and...

Hi all, I have long had this unsolved question about arclength parameterization in my head and I just can't bend my head around it. I seem not to be able to understand why velocity with arclength as the parameter is automatically a unit tangent vector. My professor proved in class that s(s) =...

Hello Forum, A conservative vector field G(x,y,z) is one that can be expressed as the gradient of a scalar field P(x,y,z). Could a time-varying vector field like D(x,y,z,t) be a conservative vector field? If not, why not? Can it be conservative (or not) at different time instants? Thanks!

Is angle a vector or a scalar quantity? (eg- θ in l=rθ)

In Latent semantic analysis, the truncated singular value decomposition (SVD) of a term-document matrix ##A_{mn}## is $$A=U_rS_rV^T_r$$ In many references including wikipedia, the new reduced document column vector in r-space is scaled by the singular value ##S## before comparing it with other...

I know that a vector quantity or simply a vector is a physical quantity which has a magnitude and is associated with some definite direction. According to this definition should not pressure and current be vectors since both are associated with some definite direction?In some places they are...

How would you write the vector loops and geometric constraints for this mechanism?