Questions on coordinate systems, a geometric method where a point in n-dimensions is assigned a corresponding n-tuple of numbers.

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Are there any differences between the mathematical definition of vectors and scalars and how they are defined in physics?

From a purely mathematical perspective, the notion of scalars and vectors and their different roles makes sense to me. Vectors are elements of a given vector space $V$, and scalars are elements of the ...
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2answers
18 views

Signs of angle measured in 3D

In a two dimensional case when angle between one of the axes and a line is to be measured, a counter clockwise rotation from the axis gives us a positive angle whereas a clockwise rotation gives us a ...
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1answer
34 views

Is this a correct question?

Q. Find the shortest distance between two non-intersecting lines passing through the points whose position vectors are a and b are parallel to vectors c and d respectively. My confusion is : two non-...
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2answers
42 views

The tangent to the curve $y=x-x^3$ at point P meets the curve again at point Q.

One point of trisection lies on $y$ axis. What is the equation of locus of other point of trisection? I have tried finding the slope, distance and tangent equation but ended up with no results and a ...
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2answers
32 views

Find angle between two coordinate systems

I have two coordinate systems (xyz and x'y'z') and I need to find the angle of rotation between the two of them. xyz is fixed and x'y'z' starts in the same position as xyz. x'y'z' is then rotated 45 ...
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3answers
46 views

Find the value of k and h [on hold]

$$ \frac{k(4-4)}{(3-1)h} =-1 $$ The numerator will become zero then LHS can never become equal to RHS. What approach will we make to solve these kind of equations and why? Exact Question- Consider ...
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3answers
37 views

What is the meaning of having a position vector $\vec r = z_1 \hat i +z_2 \hat j +z_3 \hat k$ where $z_1,z_2$ and $z_3$ are complex numbers?

What is the geometrical interpretation of this? I came across this as a solution to a second order differential equation when I was solving a problem including $m\vec a = f(\vec r)$. Thank you for ...
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3answers
88 views

Why the Jacobian isnt always 1? [on hold]

We have $A=\iint {\rm dx}' {\rm d}y'=\iint G \,{\rm d}x\,{\rm d}y$, where the integral is over a region with area $A$ in the $xy$-plane and $G$ the Jacobian of the coordinate transformation $x\to x'$ ...
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1answer
105 views

Why must a function be independent of coordinates?

What is the motivation for why a function should be independent of coordinates? In the case of a general manifold I kind of get why, since one (usually) defines a function $f$ as a map from the ...
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3answers
53 views

Problem including three circles which touch each other externally

The circles $C_{1},C_{2},C_{3}$ with radii $1,2,3$ respectively,touch each other externally. The centres of $C_{1}$ and $C_{2}$ lie on the x-axis ,while $C_{3}$ touches them from the top. Find the ...
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2answers
33 views

determine all (x,y) of the line Normal to an Ellipse

Hi everyone I have a question that requires me to determine the (x,y) coordinates of all points that intersects the x-axis on this ellipse when the normal line has a slope of -4, and I'm curious to ...
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3answers
25 views

Given the base of a triangle and the ratio of the lengths of the other two unequal sides , prove that the vertex lies on a fixed circle .

Can someone please explain to me how to approach this question ? I don't want the solutio
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2answers
36 views

Find the minimum of sum

Let $A(1, 1)$ and $B(3, 2)$ be two points. Find point $M(x, 0)$ so that the sum $AM + MB$ is minimum. After that, find the minimum of the sum. In order for the sum to be minimum, I concluded that $x$ ...
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1answer
23 views

Why Circle is traced counterclockwise and ellipse is traced clock wise?

In the Lecture 32: Polar Coordinates,professor traces the circle counterclockwise, but traces the ellipse clockwise. "Which was this one here. And first we noted that this does parameterize, as we ...
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0answers
6 views

Laplace Equation on Prolate Spheroidal Coordinates

I'm currently trying to solve Laplace's equation outside of a prolate spheroid. The scalar field has to vanish far from the spheroid. Because of the geometry I thought it might be convenient to use ...
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0answers
88 views

'Tetrahedral' coordinates in space (generalization of hexagonal coordinates)

The Cartesian coordinates are the most widely used in Euclidean space of any dimension. However, there is another set of coordinate systems which can in some way be considered optimal. Imagine ...
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3answers
101 views

Why doesn't coordinate difference between two points correspond to distance between two points?

I know that in Euclidean geometry, where the manifold is "flat" (such that it is isomorphic to an open subset of $\mathbb{R}^{n}$), $M\cong\mathbb{R}^{n}$, one can use Cartesian coordinates, $\phi (p)\...
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1answer
108 views

Connection of two objects in coordinate system

Please take a look at the following picture: I want to connect two objects by a line. This line has to start and end on an the red lines of the objects and are not allowed (at least it should be ...
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0answers
27 views

How to scale a coordinate system?

I have a bunch of coordinates and I draw them on a Bitmap Image to then display them on an Image control in a Wpf application. Given data to take into consideration: My Bitmap File is ...
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1answer
56 views

Compute the intersection of the offsets of two tangents between two circles

Pardon the amateurish notation and what's probably a very simple puzzle for most on this site. Still, the solution eludes me.... I have 3 circles $C_R, C_G, C_B$ as shown in the diagram. I know the $...
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1answer
38 views

Translating co-ordinate from one rectangle to another rectangle

I am trying to translate a co-ordinate from one rectangle to another. In realty, the second rectangle is the first rectangle "scaled-up". I am using the following method: ...
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0answers
31 views

finding farthest vertices to a point in n-dimensional hypercube

I am an engineer not a mathematician. So, please accept my apology if this is a very simple problem for this site. Given a coordinate $A(x_{1A},x_{2A},...,x_{nA})$ and a value d in n-dimension space. ...
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1answer
36 views

Understanding how to change coordinate systems

Let's say I have a canonical coordinate system in $\mathbb{R}^2$ described by the basis $\{e_1=\begin{bmatrix}0\\1\end{bmatrix}, e_2=\begin{bmatrix}1\\0\end{bmatrix}\}$. In it I have a vector $\vec{x}...
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1answer
30 views

Basis Vectors in a General Curvilinear Coordinate System

I'm confused as to how does one find out the basis vectors of a curvilinear co-ordinate system. In the context of a general, arbitrary curvilinear co-ordinate system, the textbook I'm reading states ...
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52 views

Convert velocity vector from body frame to world frame

I need to convert a 3D speed vector from a body frame to a world frame (ECEF). The velocity 3D vector I have is a linear body velocity, and I have the orientation of my object in radians in the world ...
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1answer
56 views

A new Equation of a Straight Line

Have I actually discovered a new straight line Equation? (y-y2)/(y-y1)=(x-x2)/(x-x1) is also a straight line equation?
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1answer
45 views

Necessary & Sufficient condition for the line $ax+by+c=0$ to pass through the 1st quadrant

What is the necessary and sufficient condition for the line $ax+by+c=0$, where $a,b,c$ are non-zero real numbers, to pass through the first quadrant? I could find the points at which the line ...
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4answers
69 views

Which values can $k$ take, if $(k+6)(k-2)>0$

Question: Find the set of values of $k$ for which the line $y = kx - 4$ intersects the curve $y = x^2 - 2x$ at two distinct points. I have pretty much solved this. Considering both intersect, I ...
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0answers
37 views

“A reflection across one line in the plane is, geometrically, just like a reflection across any other line.”How?

How can this statement be represented geometrically?-"A reflection across one line in the plane is, geometrically, just like a reflection across any other line." (i tried it by drawing some ...
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1answer
33 views

How to convert a 2D coordinate system into a toroidal coordinate system?

I have a 100x100 square with each point having a node attached to it such that a node at point 0,99 is more similar to a node at point 0,0 than 0,95 lets say. The same can be said about the other axis....
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2answers
10 views

Given point before and after rotation at given axis calculate the angle of rotation

I have a point at 2d space in only positive x and y axis, point P(x1, y1) is rotated along axis point C(x3, y3) to reach at point P2(x2, y2). Now I just need to calculate the angle of rotation. If ...
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1answer
25 views

Converting Ellipse Integration Boundaries To Cylindrical Coordinates

I'm having the following integral, and I'm being asked to convert the integration boundaries to cylindrical coordinates. I've figured out that on XY-plane it's an ellipse having the following ...
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0answers
25 views

Change in Coordinate System

Why when you change the coordinate system of a equation, the graph changes? For Example: $z = 10 - 3r^2$ (Cylindrical Coordinates) is a sphere. $r^2 = x^2 + y^2$ $z = 10 - 3( x^2 + y^2)$ (...
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Gaussian function at a rotated and translated coordinate system

I'm reading a paper and this coordinate transformation came along. In the $z_{i}=0$ plane the electric field is writen as $E=\exp[-x_{i}^2]$. The author says it's more convenient to work with the ...
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1answer
33 views

Mathematics of transformation of 2-D to 1-D coordination.

Let's see an example In cartesian coordinate system: ...
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0answers
41 views

Challenging calculation of a Jacobian for an unusual matrix coordinate transformation

I am studying a random matrix ensemble and I am having trouble performing a coordinate transformation. My question is very straightforward, but perhaps a bit technical. I have the following integral--...
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0answers
33 views

tangent space in a moving coordinate frame

I've got a problem in some geometry of flow. For the sake of completeness I will give the complete derivation of the equation of interest, but I will seperate it into derivation part and question ...
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1answer
32 views

Formula to map any given point on circumference of circle with given radius

I am working on a project where I need this. Mathematically : I need a formula to map any given point P(x,y) on circumference of a circle of given radius r and center c in 2D space. Insights of ...
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0answers
25 views

transform Gaussian ring from cylindrical coordinate system to Cartesian coordinate system

I have Gaussian vortex ring with the equation in cylindrical coordinate system as enter image description here with enter image description here $x_c$, $y_c$, $z_c$, $R$, $\pi$, $\gamma$ are known. ...
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1answer
24 views

Given points P (2,3),Q (4,-2),R (a,0) what should be the value of a if |PR-RQ| Is maximum?

Given points P (2,3),Q (4,-2),R (a,0) what should be the value of a if |PR-RQ| Is maximum ? I tried that maybe the points are collinear but I'm getting wrong answer applying collinearity condition i....
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1answer
20 views

how to get the angle of arc ??

dart game board is divided into sectors by 30 degrees like pizza slice. the given is (x, y) coordinates, and I need to find where coordinates are lying on. how can I get the angle just with ...
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1answer
35 views

Linear Transformation Basis Exercise

I have tried to solve the following exercise. Is it right? Consider the linear transformation L: ℝ⁴→ ℝ³. Knowing that: $$ L \begin{pmatrix}2\\0\\0\\0\end{pmatrix} = \begin{pmatrix}2\\2\\2\end{...
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1answer
24 views

Linear transformation explanation

I have the following exercise: Consider the linear transformation L: ℝ³→ ℝ². Knowing that: $$ L \begin{pmatrix}1\\1\\0\end{pmatrix} = \begin{pmatrix}1\\2\end{pmatrix} \space\space\space\space\...
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2answers
30 views

Find the nearest point

How can I find the nearest point of the set(M) to the point (A) in GENERAL(with different set or in 3D space )?
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29 views

notation of a real coordinate system of n dimension

I observe that books write a bold faced R, But the professors write R preceded by a bar. Is that I, denoting infinitism? I am not sure if it is I, its more of a vertical bar. Is it universal, or ...
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Oblate Spheroidal Coordinates, Confocal Ellipsoidal Coordinates and Geodesy

What is the name of the orthogonal coordinate system that is most commonly used in modern geodesy\geomatics engineering to model the reference ellipsoid? I suspect it is either oblate spheroidal ...
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0answers
24 views

Cartesian and Polar Coordinates, proving the same number-pair

The question states: Prove that a necessary and sufficient condition for a point to be represented by the same number-pair (a,b) both cartesian and polar coordinates is that it lies on the initial ...
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1answer
34 views

intersection of lines and circle in coordinate geometry

If line $y=kx$ divides two chords drawn from point $(\sqrt{3},1)$ on the circle $x^2+y^2=4$ in $1:2$ . Then range of values of $k$ $\bf{My\; Try::}$ Equation of chord $(1)$ which passess ...
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1answer
50 views

Algorithm for getting consecutive line segment edge points from midpoints

So I have a rectilinear grid that can be described with 2 vectors. 1 for the x-coordinates of the cell centres and one for the y-coordinates. These are just points with spacing like x spacing is 50 ...
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1answer
35 views

Help needed understanding this explanation for the Jacobian

In this Quora Answer, a very intuitive explanation for the Jacobian is provided, there is however a step I don't understand: He takes this square: And via a polar coordinates transformation ...