Questions on the use of algebraic techniques to prove geometric theorems.

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Using axis coordination to represent rotation matrix instead of angles

Euler angles give us clear matrix for conversion of a vector from car reference $Fr^C$ to earth reference $Fr^E$. If $\vec V$ is a vector in different frames it is represented differently: $$\vec V^E=...
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Analytic geometry and definite integrals problem…

So, here's the problem: We have a parabola $y^2=2px$ and a line which is perpendicular to parabola and forms the angle $\frac{3\pi}{4}$ with x axis. I have to find the area between the parabola and ...
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77 views

Easiest way to verify that $4x^2+y^2=1$ is an ellipse?

Normally I would just divide both sides by the number $4$ because it's not good in there, but I can't do it for $$4x^2+y^2=1$$ I must have $$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$ So what's the ...
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61 views

Characterize a rotation matrix

Given a matrix $A\in M_{2 \times 2}(\mathbb R)$ or $M_{3\times 3}(\mathbb R)$ how to determine if it is a rotation matrix? Is there any theorem that characterize a rotation matrix just by looking at ...
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Is there a problem in assuming that a point is the same thing of a vector?

I've read Apostol's Calculus, in the section on analytic geometry. He says that he's going to use 'vector' and 'point' interchangeably. But in Beardon's Algebra and Geometry, he argues that there is ...
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4k views

Coordinate Geometry - Area of a Quadrilateral

What is the area in square units, of a quadrilateral whose vertices are $$(5,3), (6,-4), (-3,-2), (-4,7)?$$ I have tried creating the triangles, but didn't know how to find the diagonal. I wanted to ...
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162 views

snugly fitted spheres in a cube

A larger sphere A, having a radius $R$ is snugly fitted in a cube (i.e. sphere A touches all six faces of the cube). Further, a small sphere B is snugly fitted in the corner of cube (i.e. sphere B ...
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50 views

Analytic Geometry - vectors and points

Can somebody help me? In the picture, $\|AM\|=2\|MB\|$ and $\|AN\|=\frac{1}{3}\|CN\|$. Write $X$ in function of $A, AB, AC$.
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1answer
46 views

Hyperplanes divide space

Problem. What is maximal number of connected components on which $n$ hyperplanes divides $\mathbb{R}^m$ if they all have 1 common point. In fact this problem was firstly stated in $\mathbb{R}^3$ and ...
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2answers
83 views

Surface area of a section of the unit sphere

Let $v$ be a vector on the unit sphere in $\mathbb{R}^n$ and let $S(\epsilon)$ be the set of vectors $s$ on the same sphere such that $$ |s \cdot v| \leq \epsilon.$$ What is the surface area of $S(\...
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119 views

High School Geometry problem with a triangle and trapezoid in a larger triangle.

In school, I have an assignment to write a problem for geometry students. I have written the following problem. Draw triangle ABC. Let the height have magnitude h. Draw a line segment, DE, which is ...
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94 views

Determine the value of y so that two line segments are parallel

Determine the value of $y$ so that the line segment with endpoints $P(3, y)$ and $Q(-3, -1)$ is parallel to the line segment with endpoints $R(-4, 9)$ and $S(5,6)$. I began by finding the slope of ...
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1answer
84 views

Trigonometrical functions and complex numbers

(This question will at first appear too broad. However, the overall philosophy will be explained below in a way that asks specific questions, which I hope will be conducive to this being a reasonable ...
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403 views

Rational parametrization of circle in Wikipedia

In http://en.wikipedia.org/wiki/Circle but also in the corresponding article in the German Wikipedia I find this formulation ( sorry, I exchange x and y as I am accustomed to it in this way ) : "An ...
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2answers
389 views

How to calculate the solid angle of a spherical rectangle from astronomical angles

Say I have 2 astronomical angle pairs defining a confined region on the visible hemisphere: (minAzimuth, minElevation) & (maxAzimuth, maxElevation) How can we calculate the solid angle of the ...
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48 views

The sum of the abscissae of the intersections of a cubic and a line

I remember being told in passing in a talk once the following theorem: Let $y=x^3$, and let $x_1,x_2,x_3$ be the abscissae ($x$ co-ordinates) of three distinct points on this cubic. Then $x_1+x_2+...
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36 views

Intersection of 3 positively sloped planes

Suppose I have three planes, each of which is 'positively sloped' in the sense that the first plane intersects the x-axis at a positive value, and the y and z-axes at a negative value. Similarly, the ...
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81 views

Barycentric Coordinates of Orthocenter question

this page describes the barycentric coordinates of the orthocenter as $(\tan A : \tan B : \tan C)$. How would you prove this using the areal definition of barycentric coordinates? Thank you. EDIT: ...
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74 views

Affine transformation that sends a conic to itself but does not preserve the focci or the axes [closed]

So I'm trying to find an affine transformation that sends a conic to itself but does not preserve the foci or the axes. I don't know if this can be done. I'm pretty sure that if it is possible then I ...
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1answer
21 views

Vector Calculus Question- Planes and Curves

Will you please help me in the following? Let $\pi$ be a plane perpendicular to the curve: $$ \gamma(t) = (5\cos t, 5\sin t,-2t) $$ at the point $(x(t_0), y(t_0 ) ,z(t_0)) $ . We also know the ...
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Locus of complex numbers.

Question Let $P(x,y)$ be the point on an Argand diagram representing the complex number $u=x+iy$ and satisfying the equation \begin{align*} \vert u \vert=k\vert u+a\vert, \end{align*} where $k$ is ...
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1answer
118 views

On the associative property of a binary operation of the fundamental group.

I was reading about the proof of associativity property of the operation on the fundamental group here. The book gives the following diagram then it says the reader should supply the elementary ...
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1answer
21 views

Question regarding condition of perpendicularity

Let $ax^2+2hxy+by^2=0$ be the equation of two straight lines passing through the origin. We know that the angle between these two straight lines is given by, $$\arctan \dfrac{2\sqrt{h^2-ab}}{a+b}$$ ...
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134 views

Complex hypersurface globally defined

Let $A$ be a pure one-codimensional analytic subset of a domain $D \subset \mathbb{C}^n$. Is it true that $A$ is defined by one single holomorphic equation $f(z)=0$ if $D$ is bounded and pseudo-convex ...
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27 views

Get angle in degrees of coordinate on circle.

So assume I have coordinates of two points on a circle, and the coordinate of the center of the circle. How would I go about finding the angle of the points in degrees?
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Bound for the distance of projections onto the unit sphere

Given $x \in \mathbb{R}^n$, $x \neq 0$, let $x' = x/|x|$ (where $|\cdot|$ is the euclidean length) be its projection onto the unit sphere. I would like to prove that $$ |x' - y'| \leq 2 \max(1/|x|,1/|...
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32 views

finding a point on a surface? the surface is an ellipsoid

I have drawn the cross-sections of the surface $2(x-1)^2 + (y+2)^2 +z^2 = 2$ for the given planes, but am now asked to write down a point which is on the surface. I have no idea how to go about this, ...
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31 views

Reference to line parametrization

Defining two lines in space, $\mathbb{R}^3$, as: $l_1: \textbf{a}_1+\lambda_1\textbf{b}_1$ $l_2: \textbf{a}_2+\lambda_2\textbf{b}_2$ The line to line intersection condition is: $\textbf{b}_1\cdot ((...
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1answer
31 views

Prove that $(A,B)\sim(P,Q)$ and $(C,D)\sim (P,Q)\implies (A,B)\sim (C,D)$?

I have the following laws: And I did the following: $(A,B)\sim(P,Q)\wedge (C,D)\sim (P,Q) \stackrel{?}{\implies} (A,B)\sim (C,D)$ $(A,B)\sim(P,Q)\wedge \stackrel{symmetry}{(P,Q)\sim (C,D)}\...
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How to calculate a reduced volume?

Let's say we have an irregular 3D shape with volume=V ( we know V but we don't know its equation= F). Now I want to calculate another 3D shape which is exactly the same shape but one size smaller, ...
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R. Blum equation of tangents clarification.

In Coxeter's Intro to Geometry, exercise 4 pg 114 restates a finding in Richard Blum's paper. On page 2, where he introduces the equation of the tangent lines: T(xi,eta)*T(xi0,eta0) - T^2(xi,eta | ...
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Vector on bisectrix between other two

Supose $\overrightarrow a=(2,-3,6)$ and $\overrightarrow b=(-1,2,-2)$ are represented in the same origin. Calculate the coordinates of the vector $\overrightarrow c$ that is on the bisectrix of the ...
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29 views

Find a point on the same alignment of normal vector of a plane

I need to find a point(x,y,z) that is - distance 2 from a known point P (x1,y1,z1) - on the same alignment of normal vector for plane A - P is on the plane A the same question as: Find a point ...
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52 views

Fitting a circle

Given a figure like , how can I determine the radius of the circle with middlepoint H analytically? CDFE is a square with sides 6/5, with E and F being points on the circles with radii 2.
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59 views

Prove that $|AM|=|BM|$, if …

Let $M$ be the point of intersection of the diagonal sides of a trapezoid. Let $l$ be the line through $M$ that is parallel to the bases of the trapezoid. Let $A$ and $B$ be the points in which the ...
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$\partial (Q A) = Q (\partial A$) for an orthogonal matrix $Q$?

Let $A \subset \mathbb{R}^d$ and let $Q \in \mathbb{R}^{d \times d}$ be an orthogonal matrix. For a set $B \subset \mathbb{R}^d$, denote $Q B:= \{ Qx : x \in B\}$. Does it hold for the boundary of the ...
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Help with this coordinate geometry question involving cirlces and parabolas.

Question: A point $P$ in a plane moves such that it remains at a fixed distance $r$ from a fixed point $A\equiv(r,r)$. (i) Find the equation of the locus of point $P$ (in terms of $r$). Another ...
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79 views

Algebraic calculation steps.

Can somebody explain how the coefficients $a_{11}, a_{12}, a_{22}$ are derived after rotating the ellipse below ?? $\widetilde{s_{11}} = \frac{\sum_{j=1}^n(x_{jk} - \bar{x_k})}{n}$ Thank you in ...
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Is rotation in $\mathbb{R}^d$ unique?

Let $\boldsymbol{u} \in \mathbb{R}^d$ such that $||\boldsymbol{u}||_2 = 1$ be a directional vector. Let $Q_{\boldsymbol{u}} \in \mathbb{R}^{d \times d}$ be an orthogonal matrix such that $Q_{\...
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Lattice points-Triangle

We have a triangle $ T $ with vertices at the $ \mathbb{Z} \times \mathbb{Z} $ grid . Now, consider the surface $ 2T= \{x \in \mathbb{R}^2 : \frac{x}{2} \in T \} $ ( so, double $ T $ ). Is it possible ...
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325 views

What is the reason behind the Pythagorean relation in a hyperbola?

I am currently (in my Pre-Calculus course) deriving the equations of the conic sections. I very much understand how the relationship, in an ellipse, between $a, b$, and $c$ is established. Knowing ...
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Aspect Ratio of Cylinder, Pyramid and Dome

The aspect ratio can easily be defined for rectangular geometries ($AR = height/width$). Is there a definition for aspect ratio of a dome, cylinder, and pyramid (Here standard pyramid and dome were ...
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224 views

Locus of image of point in a line.

I am given the following question: Find the locus of the image of the point $(2,3)$ in the line $$\text{L}:(2x-3y+4)+k(x-2y+3)=0$$ where $k$ is any real number. Attempt at solution. I ...
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28 views

Coordinate-geometry curiosity question

How can we draw a triangle give one of its vertex and the orthocentre and circumcentre? I tried to invoke the concept of 9 point circle and tried using the centroid but could not succeed in making ...
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1answer
516 views

Find the intersection of two lines passing through given points

Line A goes through the points (4,5) and (-2,-1) and line B goes through the points (3,3) and (6,1). At what point do they intersect? I found the equations of the 2 lines, for A I got: $y = 9-x$, ...
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60 views

How can one solve this equality geometrically?

Given that $x,y$ are real such that: $$x+2y=\dfrac{1}{2},$$ how can one show, geometrically that $$x^2+y^2\geq \dfrac{1}{20}?$$ I see that $x^2+y^2-\dfrac{1}{20}=5\left(y-\dfrac{1}{5}\right)^2$ ...
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83 views

Analytic-geometry rotation concept

I am confused how my book comes up with the following formula- Lets consider a Right angled Isoceles triangle with $2$ vertices on hypotenuse given as $(x_1,y_1)$ and $(x_2,y_2)$ Now the 3rd ...
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140 views

What are the distances from a line to the tangents of a circle?

I have a line given by two points, and a circle given by its origin and radius. I need to find the perpendicular distance between the line and the two tangents of the circle that are parallel to the ...
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1answer
231 views

Find the focus, vertex, latus rectum of the parabola

The problem is, Find the focus, equation of directrix, vertex, length of latus rectum of the parabola given by, $$\left(\alpha x+\beta y+\gamma\right)^2=Ax+By+C$$ I am stuck with the problem for ...
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490 views

Why this polynomial represents this figure?

I'm trying to understand why this figure is represented by this polynomial expression: My goal is to prove directly why cartesian product of natural numbers is equinumerous to the natural numbers. ...