Angles Definition and 911 Threads

I am trying to understand the properties of the ##SO(3)## Lie Group but when expressed via Euler angles instead of rotation matrix or quaternions. I am building an Invariant Extended Kalman Filter (IEKF), which exploits the invariance property of ##SO(3)## dynamics ##\mathbf{\dot{R}} =...

All the angles in triangle $ABC$ are less than $120^{\circ}$. Prove that $\dfrac{\cos A+\cos B+\cos C}{\sin A+\sin B+\sin C}>-\dfrac{\sqrt{3}}{3}$.

In Chapter 4, derivation 15 of Goldstein reads: "Show that the components of the angular velocity along the space set of axes are given in terms of the Euler angles by $$\omega_x = \dot{\theta} \cos \phi + \dot{\psi} \sin \theta \sin \phi, \omega_y = \dot{\theta} \sin \phi - \dot{\psi} \sin...

Hello, I didn't understand the geometry of molecules in which central atom has no lone pairs of electrons. for example, in $CH_4, NH_4^+$ molecular shape is tetrahedral and bond angle is $109.5^\circ$. How is that bond angle computed? $CH_4$ stands for liquid methane and $NH_4^+$ is a...

Now, I am sure we all know that objects and their action-reaction forces happen in terms of exerting forces at opposite directions but with the same magnitude. That said, I wish to ask: Do objects with action-reaction forces have their forces exerted in opposite angles as well as opposite...

Does the concept of the angle between two vectors make sense in Minkowski space? Does the concept of orthogonal basis for Minkowski space make sense? If it does, how is it defined? When we start with the usual (time, distance) basis for 2-D Minkowski space, the axes as drawn make a right...

1. I have calculated the first angle using Snell's Law and with subsequent proceeding angles I am uncertain whether my workings are correct since I have used basic laws of geometry that all internal angles of a triangle add up to 180 degrees and that alternate angles are equal to find proceeding...

Given that $ABC$ is an acute triangle with $AC>AB$ and $D$ and $E$ be points on side $BC$ such that $BD=CE$ and $D$ lies between $B$ and $E$. Suppose there exists a point $P$ inside the triangle $ABC$ such that $PD$ is parallel to $AE$ and $\angle BAP=\angle CAE$. Prove that $\angle ABP=\angle...

I solved for both heights: h1 = (1.85x10^4)m, h2 = (5.10 x 10^4)m Question is knowing ME = PE + KE for ball at h1 (angled 37 degrees), is it safe to assume there would be no KE as velocity would equal zero at the highest point? So the mechanical energy = mgh ? Would it be the same for ball at h2?

I have asked this question twice and each time, while the answers are OK, I am left dissatisfied. However, now I can state my question properly (due to the last few responses). Go to this page and scroll down to the matrix for sixth row of the proper Euler angles...

Summary:: We have a rotating arm, offset from the centre of rotation by a certain length, which is controlled by varying the length of a control rod. Need the angle of the rotating arm in terms of length of the rod. The blue line is a fixed column structure. CE and BD form the rotational...

Building visibiity sensor... What is best angle to detect water vapor reflection from laser, and refraction? I plan to have two open cylindrical containers painted flat black and put inside each other so the overlap is about 0.5-1" (adj to limit ambient light vs airflow). Laser is cheap red...

reducing it to various forms: for example, the one in the title, or 2*pi*k(ln m) = a(ln(n/m)), and so forth. My gut feeling is that it is true (that no such foursome exists), but manipulations have not got me anywhere. Anyone push me in the right direction? I am probably overlooking something...

In Wikipedia https://en.m.wikipedia.org/wiki/Associated_Legendre_polynomials, Section Reparameterization in terms of angles, I see this argument: Let ## x = cos\,\theta ## ## \sqrt{1 - x^2} = sin\,\theta ## This is also in Griffiths' Introduction to Quantum Mechanics. Why is this a valid...

I'm trying to find angles α and β. No additional information except: d, h, a. I already tried to figure it out by using isosceles triangles, but this is only true when there is a equilibrium of forces. I thought there are similar triangles incorporated, but I get too many unknown variables...

Hello all! Having some problems understanding this question. From what I know, bright fringes means that there will be constructive interferences, whereby ##d\sin \theta =m\lambda## In this case, I know the d (separation of silts). However, what should I put in for wavelength? I went through...

From the above two equations, I get : ##2mv = 2mv_aCos(a) + mv_bCos(b)## ##0 = 2mv_aSin(a) - mv_bSin(b)## But, I can't figure out how to prove how the angles should be. I think both the angles should be acute, but don't know how to show it.

Wasn't really sure how to ask this question since it's kind of niche. I have 3 cameras, which have the property that they occupy no physical space, and have no lens. They are defined only by the angle of their view. Now these cameras capture a certain field and take an image. The cameras will...

Hi guys, I'm trying to understand between gyroscope angular displacement and euler angles? for example { Δx = Δx + h * Rx * SCx);} this is gyroscope output about anguler displacement.This value can be used to determine angle that device created.Why we should euler angles to fly.(I know...

As indicated in the title and summary, I'm wondering if there is any large scale astronomical effort to assess directly the universal spatial expansion assumed by the Doppler interpretation of the redshift/distance relationship, by measuring individual galaxy subtended angles over time. The...

Summary: I found an old topic on this forum that describes a similar problem to what I am facing currently. I want to rotate an object with euler angle values, but the rotation has to be translated based around a specific angle. However, I do not fully understand the solution provided in that...

Hi, Could someone check that I'm right with this one, or put me right! I've worked the value out as x=30 and y (2x) = 60. I've come to this as I think it's an isosceles triangle so the base angles would be equal. Am I right? Thank you!

Hi, I'm trying to work out how I'd calculate the values in the below. Rather than just have the answer, I'd really like to understand how I'd calculate this. Thank you in advance!

Hi I am trying to understand Euler angles and why they work. What are some great books to use as a resource to build a deeper understanding of them? I want to know why 3-2-1 Euler angles are so commonly used, and proofs regarding the mathematical properties of this method..

Hi, I work as a math/science tutor, and a student had this question. He said you're supposed to find the values of x, y, and z and the only information given is that it is a parallelogram. It looks like it might be under-determined to me, every equation I write for it ends up reducing to 15x...

Hello, it is my first post here and I am not really sure whether this is the right section. I did not have the chance to take a look at the rules of the forum but I will as soon as I can (I feel a bit guilty for that, I'm sorry...). This is not "homework", I am just studying the amazing L&L...

I am simulating random angles from 0 to 2π with a uniform distribution. However, if I take the differences between random angles, I get a non-uniform (monotonically decreasing) distribution of angles. In math speek: Ai = uniform(0,2π) dA = Ai - Aj dA is not uniform. Here is a rough image of...

<Moderator's note: Moved from the welcome forum. General question on the measurement of angles and thus no homework.> So we have a physics lab and here's the diagram. I do know how to calculate all but i got negative angle. Does that mean that that angle is reference to the negative x quadrant...

Hi all, This isn't really "homework" - it's a personal project I'm working on. I'm attempting to animate some mechanical controls of a turbine engine in Adobe After Effects. Having a hard time with the math for "rotating two interconnected points". Here is a photo for visual aid: It's...

Being scratching my head for 2 days and not getting anywhere with this one. I am trying to figure out how to perform a 3D rotation described via a mix of intrinsic and extrinsic angles. Here is the problem: I have a platform in the shape of a box with sides of length lx, ly and lz. The platform...

PLEASE HELP1.What is the sum of the measures of the interior angles of a heptagon? A. 1260∘ B. 2520∘ C. 900∘ D. 1800∘ my answer is C 5.If the sum of the interior angle measures of a polygon is 3600∘, how many sides does the polygon have? A. 22 sides B. 20 sides C. 18 sides D. 10 sides MY...

I'm trying to wrap my head around the concept. we use 3 rotations to transfer our regular cartesian coordinates (3 x,y,z unit vectors) to other 3 unit vectors. each rotation is associated with an angle. so far I'm good. but now I saw in Landau's and Lifshitz's "mechanics" book this thing...

Recently I started looking back at some basic mathematical principles, and I started thinking about the theorem of corresponding angles. It's such a basic idea that it seems obvious on an intuitive level, but despite that (or possibly because of that) I can't think of a good way to formally...

Homework Statement Prove that if A,B,C, are the angles of an arbitrary triangle, then m(A)+m(B)+m(C) = 180 degrees by the following method: From any vertex draw the perpendicular to the line of the opposite side. Then use the result already known for right triangles Homework EquationsThe...

Hi there, I have another one for you (Blush) How can I efficiently determine if the angle between 2 vectors is positive or negative... Take a look at this example drawing: Known are the xy coordinates of 2 adjoining vectors, (I also have calcullated the 360 deg angle relative to the x-axis...

Hello Before I "phrase" my question (and that may be my problem), may I first state what I do know. I understand that a Rotation matrix (a member of SO(3)) has nine elements. I also understand that orthogonality imposes constraints, leaving only three free parameters (a sub-manifold) I also...

I can't see how the textbook produces the following relationships between angles: $$ \theta = \phi + \alpha \qquad (1)$$ $$ 2\theta = \alpha + \alpha ' \qquad (2)$$ My thinking is that the exterior angle theorem for triangles was used to create expression ##(1)##, but I am unsure as to how...

Hello, What is the value of x in the attached? obviously angle EAD = EDA but then what? the shape is irregular.

Hello, In the attached, what is the minimum integer value x can take? AD and BD are angle bisectors the answer is 16 - but i do not know how they did it I am totally stuck, could not think of anything here. The angle bisector formula I know does not fit here

According to Minkowski, spacetime for non-accelerated systems is a four dimensional continuum which has four axes (three space, one time) set mutually at right angles to each other. The velocity of a body moving in a straight line will be the resultant of the component velocities along the...

Homework Statement In the triangle ABC, angle A = 30 deg, angle B = 45 deg and since AC = a. The perpendicular from C on AB cuts AB in E. In this task you should calculate exact values. a) Determine AE, BC and AB expressed by a. b) Perpendicular from B on AC intersects the extension of AC in...

I am trying to work out the above problem. Known are: D,A,B,a,c,d. I need to work out b,d,G and F. Any ideas? c+d = 90 degrees. Regards, Stan

Homework Statement Two objects are kicked upward at different angles. Object A travels a greater horizontal distance than object B. Both reach the same maximum height. Which of the following statements about the objects are true? You may select more than one. 1.Object A is in the air for a...

There a relationships between angles to diagonals in a polygon?

I've attached the problem and my work. When I enter cos^1(6.890625) I get an error, but 6.9 is also not the answer and Does Not Exist is also not an acceptable answer. So where I am going wrong with this?

Homework Statement A child in danger of drowning in a river is being carried downstream by a current that has a speed of 2.35km/h. The child is 0.505km from shore and 0.780km upstream of a boat landing when a rescue boat sets out. If the boat proceeds at its maximum speed of 18.8km/h relative...

Homework Statement A particle is moving clockwise in a circle of radius 2.50m at a given instant of time. I have to find radiant and tangential acceleration and the speed of the particle. The acceleration vector is 15.0 m/s² and the angle between the radius and the acceleration vector is 30°...

Homework Statement A pilot wishes to fly at maximum speed due north. The plane can fly at 100km/h in still air. A 30km/h wind blows from the south-east. Calculate: a) The direction the plane must head to fly north. b) Its speed relative to the ground. Homework Equations Sine Rule...