Vectors Definition and 1000 Threads

Is tensor product the same as dyadic product of two vectors? And dyadic multiplication is just matrix multiplication? You have a column vector on the left and a row vector on the right and you just multiply them and that's it? We just create a matrix out of two vectors so we encode two...

I kind of just made up the questions. I realize this is a basic question but my knowledge of physics is very limited.q1 answer v_left_ball = v_left_ball - v_train v_right_ball = v_right_ball + v_train q2 answer To get the speed from Bob's frame I would use v_Bob = v_Bob + v_Alice To get the...

Summary:: summation of the components of a complex vector Hi, In my textbook I have ##\widetilde{\vec{E_t}} = (\widetilde{\vec{E_i}} \cdot \hat{e_p}) \hat{e_p}## ##\widetilde{\vec{E_t}} = \sum_j( (\widetilde{\vec{E_{ij}}} \cdot {e_{p_j}}*) \hat{e_p}## For ##\hat{e_p} = \hat{x}##...

I'm not interested in the mathematical derivation, the mathematical derivation already is based on the assumption that momentum is a vector and kinetic energy is a scalar, thus it proves nothing. Specifically, what happens if we discuss scalarized momentum? What happens if we discuss vectorized...

This is a textbook problem...the only solution given is ##3.##...with no working shown or given. My working is below; i just researched for a method on google, i need to read more in this area...use of the directional vector may seem to be a more solid approach. Ok i let ##A=(6,-4,4)##...

this is my work but the answers say 11 m/s^2 so I made an error somewhere. Also if someone could help me with solving the direction for the acceleration, that would be greatly appreciated.

In both Wald and Carroll, a type (k,l) tensor has k dual vectors and l vectors, yet a (1,0) tensor is a vector and a (0,1) tensor is a dual vector. I must be missing something simple. Please explain.

I tried splitting the forces up into F1 and F2 making Newtons second law equation into F1+F2=ma. Then I added over the the first force given. multiply the mass to the acceleration terms to get F2= (m*ai + m*aj) - F1

I have a doubt regarding the basic function of vectors and scalars in physics- What is the guarantee that every quantity(measured) in physics can be classified as either a vector or a scalar, and that while performing operations on said quantities, they will obey the already established rules...

I feel like this question is very straight forward and my explanation below summarizes the answer pretty well. Could someone confirm this or tell me if I am missing something? We have V which is a vector, but the question states it is a constant. If I take the derivative of V, represented by...

Find the question and solution here; Ok, i was able to solve this by using, ##3A=3ax+12ay+6bx+3by+3b## ##2B=2ay-4ax+4a+4bx-6by-2b## leading us to the simultaneous equation; ##7x+10y=4## ##2x+9b=-5## ##x=2## and ##y=-1## I had initially tried the approach of using ##3A=2B## →##B=1.5A## ...Then...

There is an ambiguity for me about vector components and basis vectors. I think this is how to interpret it and clear it all up but I could be wrong. I understand a vector component is not a vector itself but a scalar. Yet, we break a vector into its "components" and then add them vectorially...

##S## is a set of all vectors of form ##(x,y,z)## such that ##x=y## or ##x=z##. Can ##S## have a basis? S contains either ##(x,x,z)## type of elements or ##(x,y,x)## type of elements. Case 1: ## (x,x,z)= x(1,1,0)+z(0,0,1)## Hencr, the basis for case 1 is ##A = \{(1,1,0), (0,0,1)##\} And...

How the heck the bicycle reach from point A to point B with that velocity vector directions?

I should use the cross product but I don´t know how. I tried to calculate it but it didn´t work out as expected. Please can you give me one example how to do it ?

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 tried to find the components of the vectors. ##a_y =2.60 sin 63.0 = 2.32## and assuming the z axis would behave the same as an x-axis ##a_z =2.60 cos 63.0 = 1.18## ##b_z =1.30 sin 51.0 = 1.01## making the same assumption ##b_x =1.3 cos 51.0 = 0.82## I now think I should have switched these...

(a) I did (7.07*4.1)-(-7.03*3.94)=56.7 with this method I got this answer correct in my first attempt. (b) This where I seem to have gone wrong. I used a · b = (axbx +ayby) then I used a = sqrt(ax2+ay2) to get a single number for the answer. Filling in the numbers 7.07*-7.03 + 4.94*4.1 =...

I need help finding the resultant with vectors: 37.5N[NE] and 45N[21° S of E] I just don't know a way to find the angles within this triangle to help me get the resultant, so can anybody help me out?

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) -...

For example is this correct : 19icap.4(-i cap) = 76(i.-i)= 76 Or is it , take - out. Then -76(icap.icap)= -76 Is it -76 or 76 ?

I added x and y-axis so it would be square, and then vector bx would be same as vector a, but a didn't get it right. I am out of ideas. Can you help me?

I gathered that the final position of the vectors when added up would be (M,-M,M), but I'm not sure if this is correct.

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...

The answer in the textbook are options A, C and D. I understand why it is option A, because it is a scalar, I also get that option D is correct because the magnitude of a vector doesn't depend on the coordinate axes. I don't get how option C could be correct. If option C is correct why not D as...

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...

I'm trying to understand why it is possible to express vectors ##\mathbf{e}^i## of the dual basis in terms of the vectors ##\mathbf{e}_j## of the original basis through the dual metric tensor ##g^{ij}##, and vice versa, in these ways: ##\mathbf{e}^i=g^{ij}\mathbf{e}_j##...

Find the angle between the vectors $$v=-5\sqrt{3}i+5j$$ and $$w=5i$$

So far all I have determined is the equations of motion for the two and that is as follows. It is trivial that y(t)=v1sin(Q)t -gt^2/2 and that x(t)=v2cos(Q)t. Now the angle that is anticlockwise from the negative horizontal of the robber is 90 - Q using basic trigonometry, using this we can...

I want to find all the killing vectors of the metric ##x²dx² + xdy²##. We could guess somethings by intuition and check it, but i decided to use the equation itself. Unfortunatelly, i realized that i am not sure how to manipulate the equation $$L_{\chi}g_{ab} = g_{ad}\partial_{b}...

How to answer this question? I am working on this question. Any math help, hint or even correct answer will be accepted.

I'm tutoring an intro to meteorology pupil and learning about the conservation of potential vorticity, and realizing that I don't understand some basic rotational mechanics. For example, suppose I stand on the North Pole and hold a wheel such that the wheel's axis of rotation is parallel to the...

I am writing a code to calculate the Lie Derivatives, and so far, I have defined the Covariant derivative 1) for scalar function; $$\nabla_a\phi \equiv \partial_a\phi~~(1)$$ 2) for vectors; $$\nabla_bV^a = \partial_bV^a + \Gamma^a_{bc}V^c~~(2)$$ $$\nabla_cV_a = \partial_cV_a -...

I need to find the killing vectors of the FLRW metric. However, it seems that they are complicated. Is there a simple/general equation that gives the killing vectors for a given metric? Or do I have to solve ten independent killing equations simultaneously to find the killing vectors?

This is a question that I saw in a textbook: "If the magnitude of a+b equals the magnitude of a+c then this implies that the magnitudes of b and c are equal. Is this true or false?" The textbook says that this statement is true, but I'm inclined to believe it is false. I made a quick sketch to...

Hi, what is a unit vector? I mean, it is ##\hat{A}=\vec A/|A|##. A dimensionless vector with modulus (absolute value) one, I've read somewhere. So, dimensionless with modulus. Isn't that a contradiction? I mean, absolute value regardless dimension? Am I out of context?. ##\Bbb R^3## is a...

I attach my working below - my angle is correct according to mark scheme but magnitude isn't (should be 230). I think it's odd that my resultant velocity on a windy day is larger than velocity in still air, but apparently my angle is correct? I've been told that I've calculated the airspeed it...

Hey, so I've been studying some math on my own and I'm really confused by this one bit. I understand what contravariant components of a vector are, but I don't understand the ways in which they transform under a change of coordinate system. For instance, let's say we have two coordinate...

For car 1, the parametric equations are x = 1 + 0.8t and y=t. For car 2, the parametric equations are x=0.6s and y=2+s. (Let t and s represent time). Solving the system of equations, when the x values are equated are the y values are equated, I get s = -13 and t = -11. I assume that the 2 cars...

Let us suppose we are given two vectors ##A## and ##B##, their components ##A^{\nu}## and ##B^{\mu}##. We are also given a minkowski metric ##\eta_{\alpha \beta} = \text{diag}(-1,1,1,1)## In this case what are the a) ##A^{\nu}B^{\mu}## b) ##A^{\nu}B_{\mu}## c) ##A^{\nu}B_{\nu}## For part (a)...

Let A be invertible. Show that, if $\textbf{$v_i,v_2,v_j$}$ are linearly independent vectors, so are \textbf{$Av_1,Av_2,Av_3$} https://drive.google.com/file/d/1OuHxfUdACbpK4E5aca2oBzdaxGR0IYKv/view?usp=sharing ok I think this is the the definition we need for this practice exam question...

ok I am trying to solve some other problems following this example but can[t see how the $z_1,z_2,z_3$ are created I know it is pulled for REFF matrix

Einstein's vacuum solution metric: $$ ds^2 = -(1-\frac{2GM}{r})dt^2 +(1-\frac{2GM}{r})^{-1}dr^2+r^2 d\Omega^2 $$ which ##g_{\mu \nu}## can be read off easily. metric Killing vectors are: $$ K = \partial_t $$$$ R = \partial_\phi $$ How can I relate these to Maxwell equation?

I am reading the text 'Innovations in Maxwell's Electromagnetic Theory'. on page 44 there is a discussion on Ampere's circuital law . The passage is below. I don't understand the final statement. "In general represent a kind of relationship that obtains between certain pairs of phenomena , of...

I understand that dot product gives us a number and cross product gives a vector. Why is this vector orthogonal to the others two, and why it has magnitude |a|*|b|*sinΘ? How to use cross product? What does it give to us?

$\tiny{311.1.7.11}$ ok I am going to do several of these till I get it... Find the value(s) of h for which the vectors are linearly dependent. Justify $\left[\begin{array}{rrrrrr} 2\\-2\\4 \end{array}\right], \left[\begin{array}{rrrrrr} 4\\-6\\7 \end{array}\right], \left[\begin{array}{rrrrrr}...

ok, just now looking at some examples of how to do this $x_4$ is just a row with all zeros