Questions tagged [vectors]

Geometric object with magnitude (length) and direction.

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1answer
87 views

Is four velocity always given by $U^{\mu} = d x^{\mu}/d\tau$?

I was taught that four-velocity is defined as $${\bf U} = \frac{d \bf x}{d\tau}$$ and that it has the components $$U^{\mu} = \frac{d x^{\mu}}{d\tau}$$ where $d\bf x$ is the four displacement and $\...
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1answer
69 views

Is the normalization of the wave-function preserved due to…?

Is the preservation of the inner product the same thing as the vector length of the wave-function staying constant with it's rotation through some R2 plane (ie it's evolution through time), that is ...
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0answers
58 views

I think I broke Vectors [duplicate]

I was working on the following problem, when I ran into some trouble due to some repeated division of vectors into components. I began by drawing the forces acting on the block. I pretended that ...
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3answers
113 views

Reason behind vector addition law

What is the reason behind triangle law of vector addition, in other words, how is this really justified?
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2answers
100 views

Relationship between quantum projector operator and projection of vectors

In quantum mechanics, we define the projector operator $$\hat{P} = |\psi\rangle\langle\psi|$$ And we say that the action of this operator $\hat{P}$ over a ket $|\phi\rangle$ gives us the projection ...
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3answers
59 views

Angular momentum and angular velocity

The angular velocity $\vec{\omega}$ lies along the axis of rotation. And the angular momentum $\vec{J}$ is the cross product of $\vec{r} \times \vec{p}$. Which according to me should also lie along ...
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2answers
77 views

Identifying Vehicle Reversing

I am trying to identify if a vehicle is reversing. I have the position of the vehicle, the direction the vehicle is facing (as an angle), and the speed of the vehicle at different time points. Can ...
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3answers
213 views

Magnitude of vector quantities

The gravitational force in the vector form is defined as $$ \vec{F}=-\frac{GMm}{\boldsymbol {r^3}}\vec{r}$$ Many textbooks define its magnitude as $$F_g=-\frac{GMm}{r^2}$$ However, in the derivation ...
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3answers
1k views

Why do some people write the gravitational force as proportional to $\mathbf{r}/|\mathbf{r}|^3$?

I'm reading Mathematical Aspects of Classical and Celestial Mechanics, Second Edition by Arnold, Kozlov, and Neishtadt. It occurred to me that many people like to use third power when mention the law ...
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3answers
456 views

How to prove a 4D vector is a 4-Vector?

This is a fairly open ended question. Given a set of 4 Components, that is, a 4D Vector, what is the process for determining rather or not it is a "4-Vector" as defined in special relativity? I want ...
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5answers
83 views

Intuition of 2D Velocity Vectors [duplicate]

I understand 2D vectors in terms of displacement. For example, Bob moves 3m to the east and 4m to the north, the total change in position (displacement) equals 5m to the north east. But I don't ...
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1answer
45 views

Geometry with differential angles

In the solution to a problem, the author considers the normal force provided by an arc length of string with a differential subtended angle size, $\textrm{d}\theta$. The author reasons that this ...
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2answers
167 views

What is the meaning of transverse acceleration due to work in special relativity?

If we derive the momentum in Special Relativity we get : $$ \dfrac{\mathrm{d}\boldsymbol{p}}{\mathrm{d}t} = \dfrac{\mathrm{d}m \gamma \boldsymbol{v}}{\mathrm{d}t} = m\dot{\gamma}\boldsymbol{v}+m\...
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33 views

Order of positions of tensor/vector components in an inner/outer product

Show that if $T_i$ are the components of covariant vector T, then $S_{ij}=T_iT_j-T_jT_i$ are the components of a skew-symmetric covariant tensor S. The question is whenever working with equations of ...
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2answers
20 views

Basis of 2D motion analysis

The basic argument for analysing 2D motion is that if we have a projectile we can break its 2D motion into 2 1D motions along 2 perpendicular axes. The motions along these axes and their corresponding ...
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1answer
41 views

Appropriate Vector Diagram for Force

Which of these two vector diagrams is befitting for illustrating exerted force?
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4answers
104 views

Is my understanding of vectors correct?

I recently learned that a vector in mathematics (an element of vector space) is not necessarily a vector in physics. In physics, we also need that the components of the vector on a coordinate ...
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0answers
10 views

How do you calculate the speed of air when it fills a vacuum of 1 cm³?

At different heights, atmospheric pressure is different. If a vacuum was created of say 1 cm³, how would i calculate the velocity of air filling it back in. The adjacent layer of air would obviously ...
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1answer
26 views

Prove acceleration in orbit with newtons second law

I want to prove that the acceleration in a orbit at a given point r=(x,y) is $a=-\frac{GM}{R^3}r$ (My professor said this can be proven by newtons second law, but he never explained in detail how). I ...
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1answer
33 views

Work-Kinetic energy theorem derivation

So I came across this derivation in the book Classical Mechanics by Herbert Goldstein. I don't follow from the second step onwards. I understand that there's a dot product, but how do you compute it? ...
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1answer
34 views

Direction of cross product of two vectors [duplicate]

Are there several ways to find the direction of a×b other than right hand thumb rule?
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1answer
45 views

Question on notation for the inner product of complex vectors [duplicate]

Regarding the wiki: https://en.wikipedia.org/wiki/Sesquilinear_form#Hermitian_form you can see that the wiki states that physics defines the inner product for complex vectors as: $$\langle \, \...
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1answer
52 views

Where does 4N and 3N goes in this diagram?

Two forces of 3 N and 4 N are acting at a point such that the angle between them is 60 degrees. Find the resultant force. I successfully draw diagram for above sentence: But I am confused which one ...
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1answer
23 views

Why are the vectors canceled out in this scenario for angular momentum of a particle?

I have a study guide for our next test, and I'm trying to understand the professors answer but I don't understand why i^ * i^ = 0? Here is his work, Why do we know that the P Vector is on direction ...
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1answer
77 views

Why should Vector Components of a Weight be Perpendicular to each other?

So this question came to my head this morning. When analyzing such a problem, its easier to break the W component into 2 seperate components, here I've marked the components in Red & Pink arrows. ...
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4answers
53 views

What is the proof of parallelogram law of vector additon?

I have a question. How force can be defined as the magnitude of a ray? For instance, when we deal with two forces having magnitude 4 N and 5 N working 60 degree apart we solve the resultant by simple ...
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1answer
111 views

Why is $ |\vec F_x| + |\vec F_y| \ge |\vec F| $ for any vector $\vec{F} ?$

I know vectors additions, components, triangle inequalities, also law of cosine when length of vectors components and angle given and any math involved in it, but my question is about physical ...
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2answers
81 views

Why is work done a dot product of force and distance?

Suppose force is applied on an object at an angle theta and the block moves for some distance along x axis I understand that we take x component of ...
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5answers
848 views

Cross products/avoiding using your hand for the right hand rule in E and M

I am currently learning electromagnetism and I am getting really annoyed using a physical hand to accomplish the right hand rule. I get it, i've seen diagrams, and i've solved problems with it but ...
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2answers
49 views

Differential geometry: If $\vec v = v^i \vec e_i$, then why is $\vec r = r \vec e_r$ in spherical coordinates?

In differential geometry (and later carried over to GR) any abstract vector $\vec v$, exists on its own vector space. We can then choose to represent this vector in a coordinate basis $\vec v = v^i ...
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2answers
80 views

Do objects move in 2 directions at once?

If a velocity vector of an object can be divided into an x and y component relative to a second object's position, and both objects have gravity that attracts both objects to each other. We then know ...
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0answers
11 views

Magnitude of moment in 3d space

What's the magnitude of moment about O, if the vectors F, r1 and r2 all lie in the z-y plane? I know the magnitude of the moment is the cross product of force and the perpendicular distance from the ...
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0answers
11 views

True Airspeed from Indicated Airspeed, Temperature, and Altitude

I'm working on an aircraft problem to figure out the true heading we need to fly in order to end up where we want with the presence of wind, and I think I have the whole thing figured out except for ...
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8answers
3k views

Formal Definition of Dot Product

In most textbooks, dot product between two vectors is defined as: $$\langle x_1,x_2,x_3\rangle \cdot \langle y_1,y_2,y_3\rangle = x_1 y_1 + x_2 y_2 + x_3 y _3$$ I understand how this definition ...
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7answers
98 views

Is there centripetal acceleration with velocity of 5 m/s?

So if an object is moving at $5\ \mathrm{m\ s^{-1}}$ in a circle, is it accelerating? Me and a friend are having a conversation about this.
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4answers
425 views

Is there a scalar acceleration?

Distance is paired with Displacement and it seems to be a bigger idea than just the magnitude of Displacement. Speed is paired with Velocity. I have always thought that there is not such pairing with ...
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8answers
483 views

What actually is the vector of angular momentum?

If an object spins around a central point, it gets angular momentum which is a vector with an orientation dependent on whether its clockwise rotation or anticlockwise, i get that. But what the vector ...
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2answers
55 views

Vectors, their representation and dipole moment

I know that the length of an arrow representing a vector gives represents the magnitude of that vector, now consider a force vector. $\vec{F}$=m$\vec{a}$ , therefore in space a force vector ...
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0answers
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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0answers
35 views

What is meant by surface divergence of a vector function?

My book says: If there is a surface discontinuity in a vector field $\vec{E}$, we enclose it in a thin transitional layer (of width $h$) and apply divergence theorem. If $\hat{n}_1$ and $\hat{n}_2$ ...
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1answer
73 views

Vector calculus notation, maybe?

I just got a new book on turbomachinery that uses some notation I'm not familiar with. $$ \nabla \lor \vec{W} = -2\vec{\Omega} $$ The del-(something)-vector format makes me think its vector calculus....
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2answers
50 views

Covariant surface vector

On pg 74 of Dalarsson's Tensors, Relativity and Cosmology (The Integral theorems for tensor field chapter), the covariant surface vector was defined as: $$dS_k=\frac{1}{2}\epsilon_{kmn}dx^mdx^n=\frac{...
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2answers
79 views

There are two velocities on the sphere. How to sum them on the sphere?

As shown in the figure, there are two velocities on the sphere, one is the velocity along the meridian direction, the arc length is its size, the other is the velocity along the equatorial direction, ...
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0answers
272 views

What is an axial and polar vector?

Can someone please explain this type of vector to me, I can not understand it. Axial vectors have an inner orientation, i.e. the direction of the vector indicates the positive orientation. For ...
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1answer
31 views

Why the lens is pushed to the right after light goes through?

I am asking myself why the lens must be pushed to the right in the following scenario: (image coming from Atoms and Sporks' nice video https://www.youtube.com/watch?v=UAmdoOX3870&t=327s) This can ...
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1answer
34 views

Centripetal Force Static Friction

I understand that when a car is turning, it is using static friction to do so. The wheel is turned where the perpendicular component of the wheel is opposing the direction of motion, creating a force ...
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0answers
41 views

Transformation of Vectors

let $\Psi \in V$ be a vector and we have the action of a lorentz transformation on the object $\sigma_2 \Psi $. And $\sigma_2 \Psi $ is then in V as well. V is "Weyl or Dirac space". And the lorentz ...
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2answers
58 views

When calculating centripetal force, do we ignore non-radial or tangential forces

Suppose an object moving in circular motion in the vertical plane (ie such that gravity points directly downwards) around a central point attached by a string; the object is constantly accelerating as ...
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1answer
76 views

Can two physical vectors form a physical cross product if they are physically separated?

I would say that they can't create a cross product. If they can create a cross product, which seems to be the case from the comments below and answer, then is that cross product consider local or non-...
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2answers
78 views

Why the electric field of an infinity line is dependent on distance? [duplicate]

A very common question about Gauss law is being asked in schools or Internet: Why the electric field of an infinite plane is independent on distance? Actually I was one of who asked this ...