Questions tagged [vectors]

Geometric object with magnitude (length) and direction.

117 questions with no upvoted or accepted answers
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3-force in General Relativity

Lets consider definition of 3-force in different theories: Newtonian mechnics In Newtonian mechnics we can define the force as a cause of changing momentum $p=m\vec{v}$ with time. According to the ...
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1answer
507 views

Variables in calculation of drag coefficient

Okay, so I looked up drag force equation, and I found that the equation involved the drag coefficient. Then I looked up the drag coefficient, and the equation for it involved the drag force. ...
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78 views

Connection between contra-/covariant vectors in SR and complex numbers?

If we take a spacetime with one spatial dimension, we can write a vector as $A^\mu=(t, x)$. This is a contravariant vector, and we can calculate the covariant vector by multiplying it with the ...
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1answer
195 views

New way to compensate for gyroscopic drift

Let us assume two points in 3D space and we only care about their orientation, no velocity, position or acceleration. Now connect those two points with a rigid body of certain length, so that the ...
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138 views

Acceleration vector of tipping object (robot)

I am about to develop a simulation application of a robot. For that reason, I would need to calculate the acceleration of the robot (centre of mass) when its centre of mass exceeds its feet (ie it ...
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659 views

“Hypersurface orthogonal” component of covariant derivative of normal vector

I believe that answer to my question is rather trivial but I can't seem to get my head around it. In the context of the ADM formulation of gravity (or any other differential geometry context, I guess) ...
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1answer
176 views

Banking of road

If a car is moving on a banked frictionless road one of the component of normal reaction force acts as the centripetal force required for the turn. But normal reaction force is a reaction force. ...
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2answers
73 views

Definition of velocity in classical mechanics

Let $(r_1,r_2,r_3)$ be the coordinates of a particle $r$ in the coordinate system $\phi$. Let $\{\hat{e_1},\hat{e_2},\hat{e_3}\}$ be the coordinate basis of $\phi$. Why do we define the velocity $v$ ...
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1answer
81 views

Cross product of vectors

I am unable to comprehend the following lines given in page 657 of Shankar's Principles of Quantum mechanics: One tricky point: The cross product is defined to be orthogonal to the vectors in the ...
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3answers
103 views

How Newton's laws replicate themselves on a larger scale?

Now I was reading The Feynman Lectures on Physics and found this which I found somewhat peculiar and deep and thus want your assistance here. So here it goes: The theorem concerning the motion of ...
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1answer
99 views

Constraints on the factorization of a unit quaternion rotation operator

In Jack B. Kuipers' Quaternions and Rotation Sequences page 194-195, section 8.7.1 explains how to decompose a tracking sequence quaternion $q = q_0 + \vec{i}q_1+\vec{j}q_2+\vec{k}q_3$ into two other ...
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55 views

What are the freedom of breaking vectors into it's components?

I came across some problems in which the way I broke vectors into it's components didn't gave me answer. Here are few of them: Above mass $m$ is hanging with spring as shown, they broke $T$ in ...
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697 views

How to calculate acceleration vector of a moving object to intercept a stationary object

The short version: I'm moving on a certain straight-line trajectory (two dimensions only, thank goodness), and I want to intercept a stationary object, with constraints on maximum acceleration and ...
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534 views

Zeroth component of 4-momentum and relativistic energy-momentum relation

As I understand it one is forced to use 4-vectors since we require objects that transform as vectors under application of Lorentz transformations and 3-vectors do not (technically they do under ...
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100 views

Difference between vacuum and pseudovacuum vector?

What exactly is the difference between the vacuum and pseudovacuum vector? In my case the ground state of a system is the vacuum vector and by letting operators act on that vacuum vector magnons are ...
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1answer
315 views

Sound - for purposes of vibration

What is the best way to distribute noise from more than one source (I'm envisioning a system with many), within a dome, with the ground as its primary target, at optimal frequencies and volumes to ...
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186 views

Surface normal on the earth to the sun at a given point in time

How complicated is it to calculate a surface normal on the spherical approximation of the earths surface pointing towards the sun at a given point in time? What I try do is to highlight a small area ...
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25 views

How to solve 3-variable planar vector equilibrium problems?

While it's quite straightforward to solve problems with multiple planar vectors in equilibrium for 2 variables, I'm having issues with those where I'm asked to minimize or maximize a variable along ...
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53 views

Does the proper four-acceleration $A^{\mu} = (0,0)?$

Let the proper four-position vector $x^{\mu}(\tau) = (0, \tau)$. Differentiating this successively wrt $\tau$ I get the four-velocity $u^{\mu}(\tau) = (0, 1)$ and then the four-acceleration $A^{\mu}(\...
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40 views

Expressing Fresnel Equations in terms of $\mathbf k$-vectors

Write the Fresnel Equations in terms of the k-vectors (or the propagation constants $\gamma$). Assume that the x-direction is normal to the interface, and the z-direction is parallel to the interface, ...
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48 views

Visualizing a scenario where wave vector's direction is different from Poynting vector's one

Can someone help me visualize how the direction of the wave vector can be different from the one of the poynting vector in a scenario like a lossy medium? I'm not seeing it.
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48 views

Quick clarification of what Feyman meant here, talking about symmetry in physical law under reflection

When Feynman says "Now if the law of reflection symmetry is right in physics, then it must be true that the equations must be so designed that if we change the sign of each axial vector and each cross-...
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42 views

Rotations about a unit vector which doesn't pass through the origin in 3 dimensions

I am trying to understand how rotating a vector about an aribitrary axis which does not pass through the origin of the coordinate system $(x,y,z)$. Let the $\vec{r_{1}} $ and $\vec{r_{2}} $ be two ...
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69 views

Linear operators and the inner product

I pick the inner product involving the linear operator $\Omega$, $\langle i|\Omega|j\rangle$, from the $n\times n$ matrix $\Omega_{ij}$ as structured in page 21 of Principles of Quantum Mechanics by R....
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94 views

Invariance of the square of the four vector and a hyperbola

The invariance of the square of the relativistic four vector is given by $\begin{equation}c^2t^2-x^2-y^2-z^2=d^2\end{equation}$. What I am unsure of is if this then relates to the hyperbola shown ...
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26 views

Attitude quaternion from 2 vector measurements

I am using a method proposed by R. G. Reynolds to estimate attitude based on two vector measurements, taken from: https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19990052720.pdf Suppose we ...
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1answer
82 views

What is a vector?

"Introduction to Electrodynamics" by Griffiths has the following lines: A vector is any set of three components that transforms in the same manner as a displacement when you change coordinates. ...
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2answers
56 views

Circular area as vector

It might sound lame, but can all the area be defined as vector quantity? I understand how the area of a parallelogram or a triangle is a vector. But when it comes to a circle, I don't understand. Say ...
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3answers
82 views

Describe the dynamics of a fluctuating vector

I have a time series for the $\mathbf{v}(t) = (x,y,z)$ components of a vector quantity. It is fluctuating in time, and has a non-trivial autocorrelation function which I want to somehow elucidate. The ...
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48 views

Is it the case that vector quantities always have to do with location?

We know that vector quantities follow vector law of addition and vector addition only makes sense in a plane/space (we can treat vector addition in a line as scalar addition). It is also worth noting ...
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275 views

What physical observation led to the use of dot product and cross product of vectors in physics?

Before, I elaborate my question, I would mention that this question is similar to many questions asked on this site. Still, none of the answers satisfied me. Addition and substraction of vectors ...
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1answer
241 views

Questions regarding projecting the energy-momentum tensor for perfect fluid along the four-velocity

The energy momentum tensor for a perfect fluid is defined as: $$T^{\mu \nu} = (\rho + p)U^\mu U^\nu + p \eta^{\mu \nu}$$ Where $\rho$ is the energy density, $p$ is momentum, $U$ is the four-...
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102 views

How to write these quantum systems in vector form

I have been given these three quantum states: 1: $\frac{1}{\sqrt8}(|0\rangle +|1\rangle)(|0\rangle +|1\rangle) (|0\rangle +|1\rangle)|11\rangle$ 2: $\frac{1}{2}(|0\rangle +|1\rangle)(|0\rangle +|1\...
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171 views

Physical meaning of Vector Calculus Corollaries

The divergence theorem and the Stokes theorem have well known physical interpretations. For example, the divergence theorem merely states that the amount that a vector flows out through a closed ...
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1answer
106 views

superposition at pi/4 phase-offset of elliptically polarised light

in an earlier question, i asked if elliptically-polarised light could be superimposed in a way that allowed the vector(s) E (the jones vectors) to make sense: superposition of elliptically polarised ...
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1answer
490 views

How to calculate angular velocity about a second axis?

A point in space at $(a,b)$ revolves around the origin with an angular velocity omega. I wish to compute its angular velocity about a second axis, running parallel to the $z$ axis at point $(x,y)$. ...
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106 views

validity of “law of vector addition of current elements”

In several articles and books, I have read that "the magnetic force on or due to a small element of a circuit is equivalent to two or more of its component elements, provided that the current remains ...
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1answer
54 views

Electromagnetism Current Equation

In my book(Field and Wave Electromagnetics), it says that $ \Delta I= \Delta Q / \Delta t = \vec J \cdot \Delta \vec s $. My understanding is that $\Delta$ can be replaced by $d$ which represents the ...
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1answer
90 views

Vector Derivative: General Case

From "An Introduction to Mechanics" by Kleppner & Kolenkow, SIE-2007, Chapter 1 (Vectors and Kinematics), Section 1.8 - "More about the derivative of a vector". In this section, towards the end, ...
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59 views

Defining a room that is equally lit at every point

Imagine you were in a room designed such that no matter where in the room you stand, a sphere of volume 1 liter held at 2 meters above the ground would be exposed to the same amount of light. How ...
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132 views

Calculus/Vector Calculus and so on in special relativity book recommandation

I want to learn calculus 1,2,3 vector calculus, analysis and so on in special relativity. But I never found a good and clear book/source on it. Can someone recommend an easy book or internet source on ...
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75 views

What does 'vector-like' mean?

What are properties of vector-like field/particle? What's the counterpart of it? Chiral like?
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226 views

Why is the angular average of the direction of the momentum squared a third instead of one?

In the article http://www.nano.northwestern.edu/intranet/pubdocs/quasiclassical.pdf (in going from eq. (6.13) to eq. (6.14)) it is stated that $$\langle\hat{p}\cdot\hat{p}\rangle_{\hat{p}}=\dfrac{1}{3}...
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1answer
50 views

A way to determine if a body accelerates or loses speed at a certain time

With given vectors for acceleration and velocity, is there a way to determine if a body accelerates or decelerates at a certain time-interval? Can this be determined, for instance, by simply observing ...
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43 views

What is a diffusionless fluid?

I'm taking a course in astrophysical fluid dynamics and have come across a problem involving "small diffusionless disturbances" of a fluid. Based on the nature of the course I expect the examiner to ...
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40 views

How do I show $e_r$ and its first- and second time derivative?

What does a charge q moving in a circular orbit with a constant velocity (a "rotating charge") and a stationary observer that measures the E field look like? And how do I show $e_r$ and its first- and ...
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1answer
37 views

Laboratory fixed-vector components

What are laboratory fixed-vector components? I have an effective Hamiltonian derived from a 40-something year-old Chemical Physics paper. The article mentions the term laboratory fixed-vector ...
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1answer
84 views

Speed with wind resistance

This is probably a basic question, but it has been a while since I did anything like this. If a boat is sailing forward at speed $x$ and the direction of the wind, with magnitude $y$, is either equal, ...
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216 views

Understanding unit vectors

Trying to understand how the unit vector ${\mathcal{\hat{r}}}$ defined as $\frac{r' - r}{|r' - r|} $ (where $r'$ is the source point) works in this problem: Work out the electric field, $E$, at point ...
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234 views

Phonon vectors and characteristic length interpretation

Basically I have a set of vectors of unit length, $\{\nu_i\}$, describing the movement of phonons (all orthogonal to each other), $\{\omega_i\}$. Lets say I only have two atoms, $m_1$ and $m_2$. In ...