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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6
votes
3answers
133 views

Why is it bad to pick basis for a vector space?

Reading `This Week's Finds', http://math.ucr.edu/home/baez/week247.html, I'm informed that one should avoid picking coordinate systems and I'm unsure why that is the case. Any help on the matter is ...
1
vote
0answers
13 views

eccentricity of the conic

I'm given this question to find the eccentricity of this conic : $x^2 + ky = 0, k>0$ The given equation can be written as $x^2 = -ky$ now we can say compare this with the equation of parabola. But ...
0
votes
2answers
23 views

d3x - Cartesian to Cylindrical Coordinates

Given is $d^3x = dxdydz$ and I need to convert it to cylindrical coordinates (given through: $x = r\cos\varphi$ and $y = r\sin\varphi$). The expected result is: $(dz)(dr)(r)(d\varphi)$ and I cannot ...
0
votes
5answers
40 views

Devide line to 3 points

I have two point in a coordinates system, let's say $(x_1,y_1)$ and $(x_2,y_2)$, and I want to find the coordinates of the point that separates the line into 3 parts Like this I want to know the ...
0
votes
0answers
44 views

Vector Calculus in Curvilinear Coordinates and Index Notation

I am trying to understand how would I use the index notation in curvilinear coordinates. Checking out this reference, I got until this point $$\vec{\nabla} = \sum_a \vec{e}_ah_a^{-1}\partial_a $$ ...
0
votes
2answers
19 views

Finding the coordinates of the last point to form a parallelogram

Points: $$A(-2,6),\quad B(-5,0),\quad C(1,0).$$ Find the coordinates of $D$ such that $ABCD$ is a parallelogram. My workings: Midpoint of $AD$ = Midpoint of $AC$ Letting $D$ be $(X,Y)$ $$\left( ...
0
votes
1answer
30 views

Why is the equation for its distance from the $x$ axis is twice that of its distance from the $y$ axis $y = 2x$?

I have an understanding problem that I need to clarify.. The equation of line $L$ is $2y = -5x + 10$. A point $P$ lies on the line such that its "distance from the $x$ axis is twice that of its ...
0
votes
1answer
38 views

Calculate direction vector of azimuth and Elevation in NED

I need to calculate the unit direction vector of given azimuth and elevation angles. My target coordinate System is a NED (North east down). ...
1
vote
1answer
20 views

Rewrite equation using cylindrical and spherical coordinates.

I want to rewrite the equation $z=x^2-y^2$ using cylindrical and spherical coordinates. The cartesian coordinates are of the form $(x,y,z)$. The spherical coordinates are of the form $(\rho, \theta, ...
2
votes
1answer
28 views

Quaternion from global space to local space

I've searched but have not found a response for this question specifically. I have a smartphone with a sensor that gives me a quaternion representing its absolute rotation relatively to the following ...
2
votes
1answer
50 views

Divergence theorem in curvilinear coordinates

Suppose I have a tensor \begin{gather} \stackrel{\leftrightarrow}{A} = \begin{bmatrix} a_{11}(\vec{r}) & a_{12}(\vec{r}) & a_{13}(\vec{r})\\ a_{21}(\vec{r}) & a_{22}(\vec{r}) & ...
0
votes
1answer
39 views

How to rotate a line segment around one of the end points?

I am given x1, y1, x2, y2 and θ. How can I find x3 and y3? By the way, there can be another line segment on the other side of AB (as if the line was rotated counter-clockwise). How to find that ...
5
votes
3answers
156 views

Show that $\nabla\cdot\left(\dfrac{\mathbf{e}_r}{r^2}\right)=4\pi\delta(\mathbf{r})$ using the divergence theorem.

The book answer goes as follows: By the divergence theorem, in spherical coordinates we find ...
0
votes
1answer
17 views

Are Cartesian coordinates considered to be curvilinear coordinates?

In the wikipedia page on curvilinear coordinates it is said: "Well-known examples of curvilinear coordinate systems in three-dimensional Euclidean space (R^3) are Cartesian, cylindrical and spherical ...
0
votes
2answers
35 views

What is the locus of the points of intersection of the lines as shown in the figure?

I don't know if it should matter but the sum of the intercepts the lines make with x and y axis is constant I think. It looks like a hyperbola to me.
1
vote
2answers
39 views

Loci of intersection of lines with positive intercepts?? [closed]

How does one go about proving the equation of the curve that is formed by tracing the intersection of the lines with each other?
2
votes
2answers
40 views

How to draw a curve through every point. [closed]

I have x,y coordinates. They are arranged at fixed intervals of 1 unit along the x axis. The Y values are arbitrary. I want to draw smooth curvy line that passes through all of them. Or rather, I want ...
2
votes
1answer
66 views

Using the Dirac delta function to find the density of point masses/charges

Here is an example from a textbook: Suppose there is a unit charge or unit mass at the point $(x,y,z)=(-1,\sqrt{3},-2)$; then in rectangular coordinates, the ...
0
votes
1answer
21 views

Get two specific coordinates on a linear function.

I've been trying to build an algorithm for my Unity3D project to get some specific coordinates, but I got stuck with some math problem. I have two coordinates of ...
2
votes
2answers
47 views

Angle of rotation based on direction cosines

I have a question which is bothering me for days! Suppose that we have a fixed frame $XYZ$ and a moving frame $xyz$ in 3D. The moving frame is orthonormal and is defined based on the fixed one using 9 ...
0
votes
0answers
18 views

Relative Motion between two rotating frames

I am looking for mathematical relations between equations of motion between two rotating frames. Relating the said motion by first going to a globally fixed frame is not an issue but how to approach ...
1
vote
0answers
11 views

The existence of a specific kind of coordinate system

Suppose a massive, perfectly spherical ball is fixed on a point in space, and that there is a second free particle of negligible mass. I have noticed that the magnitude of the force exerted on this ...
0
votes
2answers
25 views

Solving vector equations of planes

Find the line of intersection of two planes denoted by: $r=\overrightarrow{b}+\lambda(\overrightarrow{b}-\overrightarrow{a})+\nu(\overrightarrow{a}+\overrightarrow{c})$ ...
0
votes
2answers
24 views

Determine whether two segments P1Q1 and P2Q2 have a common point if the (x,y) coordinates of their end points is known?

Does this question have a solution? I think it's impossible to know if line segments P1Q1 and P2Q2 intersect at all with just the information about their end points Q1 and Q2. Thanks.
0
votes
0answers
11 views

Cylindrical Coordinate System

I am to write equations of each side for a cylinder.The problem is stated below: There is a cylinder...One cuts it from somewhere near the center point ( height wise ) . So the cylinder is now split ...
0
votes
1answer
51 views

How can you find the distance between the center and edges of a rectangle - a line from centre to a edge at an angle $\theta$?

I have a case where I know the coordinates $(x,y)$ of the center of the rectangle and its edges where the line is dropped anywhere on the edges $a(x_1,y_1),b(x_2,y_1),c(x_1,y_2),d(x_2,y_2)$. Say I ...
0
votes
0answers
17 views

Cartesian coordinates conventions

Is there any historical account of how did the Cartesian coordinate system get its current conventions of orientation and representation? Are there any mathematical reasons for these conventions?
1
vote
0answers
60 views

Vector Laplacian operator in orthogonal curvilinear coordinates

I'm looking for a simple expression for the vector Laplacian $\nabla^2\mathbf{A}$ in orthogonal curvilinear coordinates. Actually, I don't require the whole thing, just the part of ...
1
vote
1answer
29 views

Matrix to represent transformation $T(a_0+a_1t+a_2t^2) = a_1+2a_{2}t$ with respect to a certain basis

I'm having so much trouble understanding this very concept.. This is a problem from the book. The Mapping $T:P_2\to P_2$ is defined by $T(a_0+a_1t+a_2t^2)=a_1+2a_2t$ is a linear transformation. ...
3
votes
0answers
21 views

Single atlas implies global normal coordinates?

Let $M$ be a class $C^k$-Riemannian manifold and suppose there exists an atlas $\langle U,\psi\rangle$ for $M$ containing only one global chart. Does this imply that the Riemmanian ...
2
votes
1answer
41 views

Integral Calculus: Plane Areas in Rectangular Coordinates

Find the area bounded by the given curves: $y^2+2x-2y-3=0$ and the $y$-axis (using horizontal & vertical strip)
1
vote
2answers
25 views

Linear Algebra Coordinate Systems Isomorphism

This is an excerpt from the book. Let $B$ be the standard basis of the space $P_3$ of polynomials; that is let $B=\{1,t,t^2,t^3\}$. A typical element $p$ of $P_3$ has the form $p(t) = a_0 + a_1t+ ...
0
votes
1answer
18 views

formula to locate x and y coordinates from two point with x and y known and distance to third point known as well

updated pictureI am trying to find the formula or at least the name of this operation to locate x and y coordinates for a point from two other reference points with x and y known and distance to third ...
2
votes
1answer
56 views

Pullback metric, coordinate vector fields..

I'm doing this computation on $\mathbb{R}^3$ with cylindrical coordinates $(r, \theta, z)$, (which aren't defined on the whole of $\mathbb{R}^3$, but I don't care about that) and I seem to get a ...
0
votes
0answers
13 views

Limiting points

For a system of coaxial circles why are there only 2 limiting points? Shouldn't there be infinite limiting points? After all system of coaxial circles are pairs of circles which have same radical ...
1
vote
0answers
27 views

Direction angle of Line segment in polar coordinates

I have a line segment given by two points $A$ and $B$, that are $(r_1,\theta_1)$ and $(r_2,\theta_2)$ in Polar coordinates. I know that the direction angle of the line segment is given by: ...
1
vote
1answer
72 views

Coordinate transformations and interpreting what the Jacobian determinant describes

Apologies for a perhaps rather trivial question, but I really want to get the concept cleared up in my head. I understand that when one changes from one coordinate system there is an appropriate ...
0
votes
1answer
46 views

Why aren't area of triangle not same when calculated by different methods in this case

I came across a question today. Two mutually perpendicular straight lines through the origin forms an isosceles triangle with the line $2x + y = 5$. Then the area of the triangle is ? I know ...
0
votes
0answers
25 views

Relationship between Normal coordinates and Spherical Coordinates

I am using the following coordinates on $S^3: (\psi, \theta, \phi)$ where $$\begin{cases}x_0 = \sin\psi,\\ x_1 = \sin\psi \cos\theta,\\ x_2 = \sin\psi \sin\theta \cos\phi,\\ x_3 = \sin\psi ...
2
votes
4answers
108 views

Calculation of $\min$ distance between $x^2+y^2=9$ and $2x^2+10y^2+6xy=1$

Calculation of $\min$ distance between $x^2+y^2=9$ and $2x^2+10y^2+6xy=1$ $\bf{My\; Try::}$ Using the fact that the distance between two curve is independent of shifting. So put $x=u+v$ and ...
1
vote
2answers
31 views

Can anyone provide a good explanation to converting a vector in cartesian coordinates to cylindricals?

I am sorry about the perhaps trivial question, but for some reason I am really struggling to do this. I was recently taught about curvilinear coordinates, which I believe provides a system for ...
1
vote
2answers
24 views

Coordinates of a matrix

So on the textbook, it gives an example: If the basis of B matrix is{$\begin{bmatrix}1&0\\ 0&0\end{bmatrix}, \begin{bmatrix}0&1\\ 0&0\end{bmatrix},\begin{bmatrix}0&0\\ ...
0
votes
1answer
17 views

Trigonometry: Find points coordinates in equally arms triangle

h have a equally arms triangle. The angle on point C is not 90 degrees. I have: The coordinates of point $C(C_x, C_y)$ The coordinates of the end point of $h$, $H(H_x, H_y)$ The length of $C$ ...
0
votes
1answer
26 views

Curve connecting two points in $\mathbb{R}^n$ passing through a hyperplane

Let $\pi$ and $\lambda$ be two distinct permutations of $1, 2, . . . , n$, and consider the points $p := (\pi(1),\pi(2), ... , \pi(n))$ and $r:= (\lambda(1), \lambda(2), ... , \lambda(n))$ in ...
1
vote
2answers
53 views

Find the equation of tangent at origin to the curve $y^2=x^2(1+x+x^2)$

How do I find the equation of tangent at $(0,0)$ to the curve $y^2=x^2(1+x+x^2)$ ? Differentiating and putting the value of $x$ and $y$ gives an indeterminate form. Can we trace the curve and ...
2
votes
0answers
44 views

Calculate the Angle between two vectors in 3d Spherical Coordinates

I have two vectors in spherical coordinates, both originating at the origin and both with the same magnitude equal to one. One is vertical: {1,0,0} and the other undefined: {Ms,Mt,Mp}. The other one ...
4
votes
1answer
49 views

Why do we need the equation of pair of straight lines?

I was studying about the straight lines in coordinate geometry and came across this topic named 'pair of straight lines'. It started in my book directly with "If we multiply the equation of two lines ...
0
votes
0answers
14 views

Translate and Rotate mesh

I have a mesh constituted of some vertices in 3d space, let's call them $(x_1,y_1,z_1),(x_2,y_2,z_2),\cdots,(x_n,y_n,z_n)$. The mesh's central point is $(0,0,0)$. How to find out the new coordinates ...
2
votes
3answers
39 views

New coordinates after clockwise rotation of triangle?

The figure below represents a triangle $PQR$ with initial coordinates of the vertices as $P(1,3)$, $Q(4,5)$ and $R(5,3.5)$. The triangle is rotated in the $X-Y$ plane about the vertex $P$ by angle ...
0
votes
0answers
22 views

Coordinates change movement problem

yesterday I asked a related question that was a specific case of my problem. It was solved here: Geometry/ Triangles problem but I would want to know if its possible to get a generic universal ...