What is Vector: Definition and 1000 Discussions

It seems to me there is a little of confusion about the definition of gradient. Take for instance a smooth function ##f## defined on a differentiable manifold. Which is actually its gradient at a given point ? Someone says gradient is the vector ##\nabla f## defined at each point, whilst...

I tried finding the resultant vector which was -6360. The magnitude of -6360 is the distance the traveler must travel to reach the start. I found the angle by using the triangular sum theorem on a right triangle that was split from a scalene triangle. The scalene triangle has side lengths of...

According to Cauchy's stress theorem, the stress vector ##\mathbf{T}^{(\mathbf{n})}## at any point P in a continuum medium associated with a plane with normal unit vector n can be expressed as a function of the stress vectors on the planes perpendicular to the coordinate axes, i.e., in terms of...

Problem: Given the line L: x = (-3, 1) + t(1,-2) find all x on L that lie 2 units from (-3, 1). I know the answer is (3 ± 2 / √5, -1 ± 4/√5) but I don't know where to start. I found that if t=2, x= (-5, 5) and the normal vector is (2, 1) but I am not sure if this information is useful or how...

These are the vector fields. I really have no idea how to see if there is a curl or not. I have been looking at the rotation of the vector fields. The fields d and e seem to have some rotation or circular paths, but I read online that curl is not about the rotation of the vector field itself...

Hello, Can someone explain how to sketch the flow profile in detail. Also, I solved for curl, but I'm getting a zero while the answer is the differentiation of the function f(y). Pls do help me out!

Goldstein 3 ed, pg 171, under" rate of change of a vector " : The author derives the relationship between the change of a vector in a stationary and rotating coordinate system. In the process he uses this assumption :>It is no loss of generality to take the space and body axes as...

Hello, guys. Interesting riddle here. I have no idea how to solve it. Tried different methods, but point is answer is always wrong, exact answer Downriver, at an angle of 53.13(degree) to the bank. That exercise is from "Pohl’s Introduction to Physics"

chapter 4.8 of Goldstein’s classical mechanics 3rd edition that deals with infinitesimal rotations, and the following is the part I got stuck: (p.166~167) : I'm not able to understand what the author is trying to say. How does "If ##d\boldsymbol{\Omega}## is to be a vector in the same sense...

so my question is a little dumb, if an object on a scale has a weight that points downwards and the scale exert normal force on the object upwards cancelling the forces acted on the object then what does the scale read? also the action-reaction pair to the normal force is supposed to be the...

I really don't know how to proceed if I'm not using an specific coordinate system, Is there a way of doing this using only indices, in general form?

So when evaluation the cross product of the velocity of the charge and the unit vectors associated with the point I am getting v x r = j x [ i + j]. Well j x j is 0. j x i = -k, but yet the answer is positive. Why is this?

I used the above equation, and started with getting the cross product of dl and r, which was equal to 0.00195i+0.00365k. From there, I divided each component by the magnitude of radius cubed (0.827^3). I then multiplied by I and u naught(u_0=4pi*10^-7), and then divided by 4pi. The answer I got...

With no applied moments, it is asked to prove that a gyroscope Fermi-Walker transports its spin vector ##S_{\alpha} = - \dfrac{1}{2} \epsilon_{\alpha \beta \gamma \delta} J^{\beta \gamma} u^{\delta}##. In a local inertial frame ##u^{\alpha} = (1, \mathbf{0}) = \delta^{\alpha}_0## and...

Hi everyone! I'm having some difficulty showing that the variation of the four-velocity, Uμ=dxμ/dτ with respect the metric tensor gαβ is δUμ=1/2 UμδgαβUαUβ Does anyone have any suggestion? Cheers, Rafael. PD: Thanks in advances for your answers; this is my first post! I think ill be...

This isn’t a HW problem per say, but it’s an example shown in my statics textbook that is used to try an explain that you need to add a couple moment to move a force to a point not on the line of action, and I’m just not seeing how the direction of the couple moment is correct. See the image...

I'm preparing for exam but it seems I can't find problems similar to this on the internet. Here I will apply Gauss's law on the electric field vector to get the charge density. but the problem is that I can't find similar examples on the internet that uses direct vectors on Maxwell's equations...

Hi Here is my attempt at a solution for problems 1) and 2) that can be found within the summary. Problem 1) a = 3-2i b= -6-4i c= 4+ 6i d= -4+3i Now, to calculate each vector modulus, I applied the following formula: $$\left| Vector modulus \right| = \sqrt {(a^2 + b^2) }$$ where a = real part...

I actually have worked through the solution just fine by taking the derivative of \vec{L}: \frac{d \vec{L}}{dt} = \dot{\vec{v}} \times \vec{M} - \alpha \left(\frac{\vec{v}}{r} - \frac{\left(\vec{v} \cdot \vec{r}\right)\vec{r}}{r^{3}}\right) I permuted the double cross product: \dot{\vec{v}}...

My solution for the vector potential ##A=2Cln\frac{x^2+y^2}{z^2} \hat{z}## is: a) I used the following formula to calculate the magnetic field $$ \mathbf{B} = \nabla \times \mathbf{A} = \left( \frac{dA_z}{dy} - 0 \right) \hat{x} + \left( 0 - \frac{dA_z}{dx} \right)\hat{y} + 0 \hat{z} =...

I have a quick question. If a Vector is contained inside a plane, would the normal of the plane be orthogonal to said vector? Thank you!

Summary:: I need to solve a problem for an assignment but just couldn't find the right approach. I fail to eliminate b or c to get only the magnitude of a. Let a, b and c be unit vectors such that a⋅b=1/4, b⋅c=1/7 and a⋅c=1/8. Evaluate (write in the exact form): - ||4a|| - 3a.5b - a.(b-c) -...

The vector equation is ## v(x)=(e^x cos(2x), e^x sin(2x), e^x) ## I know the arc-length formula is ## S=\int_a^b \|v(x)\| \,dx ## I found the derivative from a previous question dealing with this same function, but the when I plug it into the arc-length function I get an integral that I've...

I have video data that shows an object moving up and down. I'd like to extract the frequency the object moves. Following the given example here (scroll down to "Examples"), am I correct in assuming Fs would be camera frame rate and L would be the total number of frames? Thanks so much!

If the energy is a vector, which as i understand for example, the potential energy , U=mgh, where g is the gravitational force, Then U is the product of scalars and vectors, so its a vector In that case being E a vector , can it be equal to mc2 (each are scalars). Like mulitplication of scalars...

At the moment he wrote that ##\frac{1}{2}mv_2^2=\frac{1}{2}m(-\dot{y}+\dot{x})^2## But, I know from vector ##v_2=\sqrt{(-\dot{y})^2+(\dot{x})^2}##. At first I (he) found that ##v_2=-\dot{y}+\dot{x}##. But, when thinking of simple velocity in ##x## and ##y## coordinate then I get...

I had an equation. $$T=\frac{1}{2}m[\dot{x}^2+(r\dot{\theta})^2]$$ Then, they wrote that $$\mathrm dr=\hat r \mathrm dr + r \hat \theta \mathrm d \theta + \hat k \mathrm dz$$ I was thinking how they had derived it. The equation is looking like, they had differentiate "something". Is it just an...

I am following along with an example in my book regarding force from an electric charge. I understand the process but I believe I am getting something wrong when it comes to adding the vectors. Essentially, F13 is equal to -1.35*10^-3 j and when I add that to the j component of F23 which is...

The picture may be blurry. I couldn't take more less blurry picture hence, giving it. The question is : Find value of ##\theta## when ##V_x## component and ##V_y## component same. I was using a simple equation of vector. $$C=\sqrt{A^2+B^2+2AB\cos\theta}$$...

Is the following true? ##\left| \frac{d\vec{u}}{d t} \right| \overset{?}{=} \frac{d |\vec{u}|}{d |t|}##

The unit vector r roof points in the direction of increasing r with phi fixed; phi roof points in the direction of increasing phi with r fixed. Unlike x roof, the vectors r roof and phi roof change as the position vector r moves. What I was thinking of the image is Although, I was thinking why...

My notes says that the geometrical meaning of $$|\vec v \times \vec w | $$ is the perpendicular distance from point ##V## to line passing through ##O## and ##W## (all vectors are position vectors) $$|\vec v \times \vec w | = |\vec v| |\vec w| \sin \theta$$ From the picture, the perpendicular...

This is the problem, I managed to solve it, i just want to check if there is an alternative approach. Find my solution below; ##\vec x= -\vec a-\vec b-\vec y## ##\vec y= -\vec d+\vec c-\vec b## therefore, ##\vec x= -\vec a-\vec b+\vec d-\vec c+\vec b## ##\vec x= -\vec a+\vec d-\vec c##

Question:

Find a vector that is perpendicular to the plane passing through the points P (1, 2, 3), Q (2, 3, 1), and R (3, 1, 2).

Is the following a correct demonstration that quantum mechanics can be done in a real vector space? If you simply stack the entries of density matrices into a column vector, then the expression ##\textrm{Tr}(AB^\dagger)## is the same as the dot product in a complex vector space (Frobenius inner...

It's a puzzle. I have decomposed vector v by using formulas known from physics: m*g*sin(theta) and m*g*cos(theta). I got: ##\vec v = (5, 5*\sqrt{3})## But it has been marked as wrong. Consequently, the rest of my calculations is not correct. Could you tell me, why?

Error Vector Magnitude is widely used in the telecommunication industry to assess the performance of the users. In the given formula, Sr(n) is the received symbol and St(n) is the ideal symbol. N is the total number of symbols received. I have a Multi-User MIMO system simulation where there are...

It is said that: It is not possible to write a position vector in a curved space time. What is the reason? How can one describe a general vector in a curved space time? Can you please suggest a good textbook or an article which explains this aspect?

Hello, I have a question regarding the contravarient transformation of vectors. So the formula: V'n = dx'n / dxm Vm So in words, the nth basis vector in the ' frame of reference over the mth (where m is the summation term) basis vector in the original frame of reference times the mth...

Velocity is a 4-vector which has 3 space dimensions and 1 time dimension. It's space parts will be directed at the 3 space directions and time parts will be directed at the time dimension (But it is inverse. So, will it point at the inverse direction?). How can someone Visualize it? How they...

Could someone explain the green highlight to me, please?

Apostol defines limit for vector fields as > ##\quad \lim _{x \rightarrow a} f(x)=b \quad(\rm or\; f(x) \rightarrow b## as ##x \rightarrow a)## means that : ##\lim _{\|x-a\| \rightarrow 0}\|f(x)-b\|=0## Can't we say it's equivalent to ##\lim _{x \rightarrow a}(f(x)-b)=0##

For a massless particle let\begin{align*} S[x,e] = \dfrac{1}{2} \int d\lambda e^{-1} \dot{x}^{\mu} \dot{x}^{\nu} g_{\mu \nu}(x) \end{align*}Let ##\xi## be a conformal Killing vector of ##ds^2##, then under a transformation ##x^{\mu} \rightarrow x^{\mu} + \alpha \xi^{\mu}## and ##e \rightarrow e...

As an aside, fresh_42 commented and I made an error in my post that is now fixed. His comment, below, is not valid (my fault), in that THIS post is now fixed.Assume s and w are components of vectors, both in the same frame Assume S and W are skew symmetric matrices formed from the vector...

Hi, A pinball moving in a plane with velocity s bounces (in a purely elastic impact) from a baffle whose endpoints are p and q. What is the velocity vector after the bounce? I don't understand how to answer this question? Any math help, hint or even correct answer will be accepted?

Given the equation ##\frac{xy} 3##. It is a fact that the gradient vector function is always perpendicular to the contour graph of the origional function. However it is not so evident in the plot above. Any thought will be appreciated.

We have a scalar potential $$\Phi(\vec{r})=\frac{q}{4\pi\epsilon_0} \left( \frac{1}{r} - \frac{a^2\gamma e^{-\gamma t}\cos\theta}{r^3}\right)$$ and a vector potential $$\vec{A}(\vec{r})=\frac{a^2qe^{-\gamma t}}{4\pi\epsilon_0r^4}\left(3\cos\theta\hat{r} + \sin\theta\hat{\theta} \right) .$$ how...

Solution: u = [-2,3,1] Po = (6,0,0) & P = (4,2,3) PoP = v = [-2,2,3] Therefore, the answer is [6,0,0] + r[-2,3,1] + q[-2,2,3]; r, q are real numbers I don't understand why (6,0,0) is used as the point in the vector equation, since it only lies on the [-2,2,3] vector, not the u = [-2,3,1]...