# What is Points: Definition and 1000 Discussions

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1. ### Critical points of matrix

My attempt is: Condition for critical point is ##x' = y' = 0##, ##0 = x - 2y \implies 2y = x## ##-2x + dy = 0## Then ##-4y + 4y = 0## However, this means that critical points are ##(2y, y)## as system is linearly dependent (both equations are the same) where ##y \in \mathbb{R}##. However, that...
2. ### I Optimal space between anchor points on a long piece of material

Let's say I want to fix a 4x4 lumber against a flat board, or any other material against any other. Depending on the situation, I may use 3, 4 or 5 screws or even more. Regarding the last and first screws, what distance should there be between them and the end of the lumber to ensure an even...
3. ### Complex mapping z ↦ ω = (z − a)/(z − b)

All points on that line are equidistant from the points a and b. Thus, the length of ##\frac {z - a} {z - b}## is 1, i.e., the points on the unit circle. If the angle of ##z - a## is ##\alpha##, and the angle of ##z - b## is ##\beta##, then the angle of ##\frac {z - a} {z - b}## is ##\alpha -...
4. ### Describe the set of points that satisfy ##(x-4)(z-2)=0##

We are told that ##(x-4)(z-2)=0## The ##0## refers to what variable? The ##z=2## i think the sketch on diagram may be misleading (not to scale). Why have the ##x## crossing the axis on negative side and same applies to ##z##... To put this into context, we may have ##(x,y,z)=(4,y,z)## where...
5. ### Potential difference between 2 points in a capacitor circuit

In the given circuit, a transient current will flow and when this current finally stops at equilibrium, the charges ##q_1## and ##q_2## are assumed to deposit at the capacitor plates as shown below. The dashed line indicates an isolated system that will have it's total charge conserved. If I...
6. ### Number of lines equidistant from four points on a plane

Hi, i'm trying to solve this problem. It's exercise 3 on page 5 from this book: Challenging mathematical problem with elementary solutions The solution is on page 41: I'm OK with the 4 circles in case 1: i can pick (inside/outside): ABC + D, ABD + C, ADC + B, BCD + A. What i cannot...
7. ### Plotting points in three-dimensional space

For this problem, , The four points are, ##P(8,2,6)## ##R(-2,16,-2)## ##Q(3.9,2)## ##S(\frac{14}{3}, \frac{20}{3}, \frac{10}{3})## And the solution is, However, does someone please know what in the proportion 2:1:3 mean? Many thanks!
8. ### A Calculating Probability of N Points on a Line Being Within Given Distance

Probability of any random n points on a line being within a given distance Hi, I am a software engineer trying to solve the following problem analytically given a line segment in cm and n random points on it what is the probability that the distance between any 2 consecutive points on the...
9. ### I Realistic steering wheel angle when car follows points along a path

Good morning , I managed to simplify the physics of my car with only the centrifugal and traction forces. If I'm in control of it, the realism is acceptable for a city builder. On the other hand, I encounter a problem when the car follows an array of points. The path of the car on my graph...
10. ### B Arithmetic mean of an infinite number of points?

So I was thinking about arithmetic, geometric and harmonic means when I had a thought. Let's say we have a curve y = x^2. We want to find the AM of the points on the curve between x=1 and x=2 i.e. y = 1 and y = 4. To make thing easier, we'll start with just the endpoints and keep adding...
11. ### I Finite many Lattice Points in Sphere?

Hello, I am wondering if in an n-ball the number of lattice points is finite. First, we have a ball which is bounded by the radius. The distance between two lattice points is given by the successive minimum. Theoretically, one could now draw a ball* around each lattice point in the (big)...
12. ### B How is the harmonic mean affected by additional data points?

We have a collection of 8 discrete data points. They are: 10, 20, 30, 20, 30, 40, 30, 40 In increasing order: 10, 20*2, 30*3, 40*2 The harmonic mean of this data series is 22.86 I read on Wikipedia that the harmonic mean is skewed towards the smaller values i.e. smaller values will affect the...
13. ### Book recommendations about singular points of algebraic curves

I'm not quite sure if this is an appropriate question in this forum, but here is the situation. I have just finished my graduate studies. Now, I want to explore algebraic geometry. Precisely, I am interested in the following topics: Singular points of algebraic curves; General methods employed...
14. ### I Prove that a triangle with lattice points cannot be equilateral

I assumed three points for a triangle P1 = (a, c), P2 = (c, d), P3 = (b, e) and of course: a, b, c, d, e∈Z Using the distance formula between each of the points and setting them equal: \sqrt { (b - a)^2 + (e - d)^2 } = \sqrt { (c - a)^2 + (d - d)^2 } = \sqrt { (b - c)^2 + (e - d)^2 }(e+d)2 =...
15. ### Algorithm problem involving 3 points and 3 lines in the x,y plane

1- Coordinates of two points are given in x and y plane. A(x1,y1), B(x2,y2) Calculate the angle between the two lines passing through each of these points with the origin of linear coordinates. 2- If a line passes between the two points A and B above, does point C lie on this line? C(x3,y3) how...
16. ### Simple Linkage with Reciprocating points

TL;DR Summary: Anybody help me get my head around this please Hi, If anybody could give me a steer on solving this that would be great. I am trying to solve the maximum forces at points fA and fB If the force from the drive TA = 64/0.6 = 1066.666N Point TB has equal arms so the same force...
17. ### Points of inflection on acceleration graph

EDIT: For this part(b) of this problem, The solution is However, isn't there more points of inflection than just ##t = 3,5 s ##? Points of inflection is when ##x'' = a = 0## so it should be ## 3 ≤ t ≤ 5 s## I also have a question about part(d): The solution is However, could I tried...
18. ### Finding charge on a capacitor given potential difference across two points

For this part(b) of this problem, The solution is However, I tried solving (b) like this: Since ##Q_{total} = 363 \times 10^{-6} C## then ##Q_1 = 181.5 \times 10^{-6} C ## since the equivalent upper capacitor is in series with the equivalent bottom capacitor so should store the same amount...
19. ### Interpolate between 2 impact points only given the throw angles

Top-Down-Perspective: At first I am quite sure that the problem is not solvable since there are that many unknowns. But my Approach would be to create a linear function with P1 and P2 and then set it equal to the function that gets me the impact location of P3 and then solve it by b3. Thanks...
20. ### B Distance b/w 2 Points on Spacetime Diagrams: Meaning?

With one spatial dimension ## x ##, the spacetime interval between two events ## A ## and ## B ## is ## \Delta s= \sqrt{(ct_{B}-ct_{A})^{2}-(x_{B}-x_{A})^{2}} ## I have a technical question: on the plane of the graph (x,y=ct), the ordinary distance between 2 points ## A ## and ## B ## is ##...
21. ### Tetrahedron with 3 points fixed, and force applied to 4th

My approach to this problem is to recognize that the tetrahedron being still means that net torque is zero and net force is zero. Fd is given Fa + Fb + Fc = -Fd Fa X a + Fb X b + Fc X c = <0,0,0> This can be split up into a series of 6 equations, 2 for each component. However, this is where I...

23. ### Find the two points on the curve that share a tangent line

IMPORTANT: NO CALCULATORS I assumed two points, (a, f(a)) and (b, f(b)) where b is greater than a. Since the tangent line is shared, I did f'(a) = f'(b): 1) 4a^3 - 4a - 1 = 4b^3 - 4b - 1 2) 4a^3 - 4a = 4b^3 - 4b 3) 4(a^3 - a) = 4(b^3 - b) 4) a^3 - a = b^3 - b 5) a^3 - b^3 = a - b 6) (a...

25. ### I Determining the Number of Points in the n-Space

Electron gas is a collection of non - interacting electrons. If these electrons are confined to certain volume (for example, cube of metal), their behavior can be described by the wavefunction which is a solution to the particle in a box problem in quantum mechanics. Allowed energy states for...
26. ### A Connectedness of Boundary Points in Schwarzschild Penrose Chart

In the Penrose chart for Schwarzschild spacetime, the boundary "at infinity" appears to be connected all the way around. I want to explore what that means physically and whether particular boundary points that appear to be connected on the chart actually are. I am using the following notes as a...
27. ### Creating a grid type 3D data array from data points

I have a 3 data column ##(X, Y, Z)## ranges from ##(min, max)##. For example, ##X = (0, 5)##, ##Y=(0, 3)##, ##Z=(0, 2)##. By using them I need to create a numpy array in the form of ##[(0, 0, 0), (0, 0, 1), (0, 0, 2), (0, 1, 0), (0, 1, 1), (0, 1, 2), (0, 2, 0)...]## So in total there will be...

29. ### Calculate tilt angle of a bar lifted via two ropes on fixed points

Summary: How to calculate the equilibrium angle of a bar that is lifted on its two ends with ropes attached to fixed lifting points? Hello and good day all, First of all I would like to apologize in advance for my english, I am not a native speaker so some grammar errors may be left. My...
30. ### A Mapping High Symmetry Points from Primitive to Conventional Cell

We usually plot electronic bands with the help of high symmetry points of the irreducible zone of primitive cell of particular material. But if we want to plot bands with conventional cell, we have to map the high symmetry points from primitive cell to conventional cell. so how can we map the...
31. ### Three Points at Vertices of Equilateral Triangle

I’m not sure of how to begin solving this problem. I attempted to draw a diagram and finding the components velocity of each initial velocity vector but this did not lead anywhere. Could so please have a hint?
32. ### Find how many points on a circle have an integer distance from other points

Distance between point (-4, 5) and point on circle: $$d=\sqrt{(x+4)^2+(y-5)^2}$$ $$=\sqrt{x^2+8x+16+y^2-10y+25}$$ Then substitute ##y^2## from equation of circle: $$d=\sqrt{x^2+8x+16-x^2+4x-6y+12-10y+25}$$ $$=\sqrt{12x-16y+53}$$ After this, I need to try the points one by one to check whether...
33. ### 2 Cords affixed at different points are wrapped on a disc

The hint says the following: "Since the cords are inextensible, every particle of a cord must be in circular motion about the point where it is affixed to the ceiling. Therefore, the velocities of the points where the cords are leaving the disc are perpendicular to the string" Due to the fact...
34. F

### Stationary points classification using definiteness of the Lagrangian

Hello, I am using the Lagrange multipliers method to find the extremums of ##f(x,y)## subjected to the constraint ##g(x,y)##, an ellipse. So far, I have successfully identified several triplets ##(x^∗,y^∗,λ^∗)## such that each triplet is a stationary point for the Lagrangian: ##\nabla...
35. ### MHB Calculate Data Points to Match Given Totals

Hello! I have a super tricky problem for everyone I truly hope there is an answer to this. I have different data sets all of which are a different amount of number per set (total points per set) Each number is multiplied by 4.86 then rounded down. Then the total is added Alternatively the sum...
36. ### Rotate Points on Sphere by Theta and Phi

Dear Forum, My goal is to rotate several points on a sphere by a theta and phi. For example, I have a sphere where the elevation is theta (90 to -90) and the azimuthal is phi (-180 to 180). I have the following points on the sphere: theta = [45 45 45 45] phi = [-180 90 90 180] This generate...
37. ### B: Calculate the distance between two points without using a coordinate system

Dear all, the following problem is not a home-work problem. I have come up with this question for myself. Nevertheless, I am stuck and need your help. The question is: Can I calculate the distance between points A and B from this information? And if yes, how? I think it should be possible...
38. ### B Distance between two points on Earth doesn't increase with altitude?

Consider the following example: Point A has coordinates 45 lat, 0 long. Point B has coordinates 45 lat, 2 long. Both points are 5000 ft above sea level. The distance between them is X. Point C has coordinates 45 lat, 100 long. Point D has coordinates 45 lat, 102 long. Both points are at sea...
39. ### Writing: Input Wanted Different Cosmic Disasters for Different Plot Points

Hi again! For my story about the generation ship “Exodus”, I “need” different kinds of cosmic disasters. Some of them actually occur within the story itself; for others, it will just be the impending danger of such a disaster that motivates people’s actions.1) There needs to be a reason for why...
40. ### Help with the Key Points of Zener Breakdown and Avalanche Breakdown

Good day, all I am familiar with both of terms that I speak of in title. But I cannot find a full answer, so I might as well ask the PhD'ers here. What is really happening in Zener and Avalanche breakdown? I have read Guide to Zener Effect and Avalanche Effect and still feel confused. And yes I...
41. ### Voltage Null Points of a Mechanical Meter Bridge

My reasoning is that the wire has a resistance/unit length. The resistance of AD must equal that of AB for D to be called a null point. If we move jockey to the left, the resistance of AD decreases, thereby decreasing the potential drop across it. This means that the V of D is more than that of...
42. ### B Questions about Lagrange points in the Pluto / Charon system

Hello. With the recent interest in the JWST orbiting at the L2 Lagrange point of the Earth - Moon system, I was wondering about the dynamics of the Pluto - Charon system. Specifically, the barycentre of that system. This barycentre lies at a point in space between these two bodies. Does...
43. ### Direction of motion of points on a rope as a wave travels

The second diagram is my attempt at the solution, in which the dotted part is the pulse in the rope a very small interval of time after ##t=0##. Point A should be at rest since we know wave is moving towards right and point A on the rope becomes a part of initial horizontal part of the string a...
44. ### Gallium Alloys and their melting points?

I am looking to try to create an alloy of gallium (probably with Tin) with a goal of increasing the melting point from 29°C to around 40°C. My current theory is to simply try to average the two melting points out by mass, which my calculations imply means 95% Gallium to 5% tin. my calculation...
45. ### I A curve that does not meet rational points

This is just to recall a nice fact: Any two points ##A,B\in\mathbb{R}^n\backslash\mathbb{Q}^n,\quad n>1## can be connected with a ##C^\infty##-smooth curve that does not intersect ##\mathbb{Q}^n##. The proof is surprisingly simple: see the attachment
46. ### Points of contact and contaminations

My small blanket fell into the floor with a dry floor mat on it. I'd like to know what kind of contamination can occur here. That is, do only the dusts in the dry floor mat transfer to the blanket? Can the dusts be removed just by moving the blanket fast up and down? Or could there be bacterial...
47. ### Car lift using suspensions as lift points

Some car lift service centers want to lift up the car by using the suspensions. I always told them to lift by the pinch weld jack points near the tires. Are there some suspension system where the service center can lift the entire car by via the suspensions. And are there some cars which they can't?
48. ### Wheatstone bridge and conditions for current to flow between 2 points

Hello, I was reflecting on the functioning of a balanced Wheatstone bridge circuit: Point C and D are at the same electric potential V (zero potential difference). Even if they are connected by an ideal zero resistance wire, current will not split at either node C or D to flow through that...
49. ### Potential difference between two points in an electric field

So I have been given a uniform electric field ##\vec{E}=20 V/m## in the direction as show in the image. I have been told to calculate the potential difference ##VC - VA##. According to the teacher (on YouTube) the potential difference ##VC - VA = -10\sqrt{2}V##. But I say it's ##-20 V## as...
50. ### Understanding Fixed Points in Hamiltonian Systems

Hi, I was attempting a question about Hamiltonian systems from dynamic systems and wanted to ask a question that arose from it. Homework Question: Given the system below: \dot x_1 = x_2 \dot x_2 = x_1 - x_1 ^4 (a) Prove that the system is a Hamiltonian function and find the potential...