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

learn more… | top users | synonyms (1)

2
votes
2answers
131 views

The Efficiency of Random Parking Problem

A few days ago in my Calculus BC class we were given a page of 6 challenging end of the year problems. That was a refreshing change from the drudgery we usually do (WebAssign). One of them went like ...
1
vote
2answers
23 views

Find the slope of a line $L$ that tangent to the graph of $y = x^3$ and passes through the point $(0,2000)$.

Find the slope of a line $L$ that tangent to the graph of $y = x^3$ and passes through the point $(0,2000)$. Well, I am new to this concept, to me, slope means $\dfrac{dy}{dx}$, but I get ...
0
votes
1answer
48 views

% of traversal for a point between two other points along a line

I'd like to solve for some "phase" or percentage, involving an arbitrary location (xC, yC) between two points. I'm not familiar with how to phrase this question, so please excuse my ignorance. Not ...
3
votes
1answer
112 views

Largest of the smallest angles of incidence from arbitrary point to tetrahedron vertex/centroid line

Picture a regular tetrahedron where each vertex has a line through the centroid and a plane normal to it. I need to show that the range of the smallest angles of incidence from an arbitrary point to ...
1
vote
2answers
59 views

Let $y=x^2+ax+b$ cuts the coordinate axes at three distinct points. Show that the circle passing through these 3 points also passes through $(0,1)$.

Let $y=x^2+ax+b$ be a parabola that cuts the coordinate axes at three distinct points. Show that the circle passing through these three points also passes through $(0,1)$. Since, the graph of the ...
1
vote
1answer
39 views

Conjugate Hyperbolas.

What would be a good approach to tackle this problem. In a previous assignment I managed to show Pq=Pr. How do I show that this tangent intersects the conjugate hyperbola. Should I start by ...
0
votes
1answer
26 views

Analytic geometry, distances

Find the equation of the geometric place: Whose distance to the point $(4,0)$ equals half the distance to the straight line $x=19$ Im using the formula for distance between points $P(4,0), Q(19,0)$ ...
2
votes
4answers
35 views

Finding horizontal tangents to a function.

Find the points at which the line tangent to the following function is horizontal $$q(x)=(x+3)^4(2x-1)^7$$ Every time I've gotten to the point of finding $x$ the numbers are all irrationally too ...
0
votes
0answers
34 views

Reference request: history of analytic geometry

I am searching a book in the domain of the history of math, that describes the historical origins of analytic geometry, starting from Descartes (?), and that describes also its development (e.g. the ...
0
votes
0answers
11 views

analytic geometry question involving perpendicular vectors

Determine the parametrics equations of the straight line that passes by $A(-1,4,5)$ and is perpendicular to $r:P=(-2,1,1)+t(1,-1,1)$. Someone can solve this? I'm trying for more than a hour and I'm ...
0
votes
0answers
25 views

Intersection of a curve with a complex line

Given: $$ \left\{\begin{matrix}t =\frac{1}{n}\sqrt{n^{4}-z^{2} } & \\ z=im & \end{matrix}\right.$$ with $n<m$, positive integers (and $i$ the imaginary unit), if one wanted to ...
1
vote
1answer
31 views

Definition of angle between non-differentiable curves

(Background: I am trying to understand the definition of angle-preserving function..I posted a question earlier but I still have doubts) My question is:how is the angle between two curves defined if ...
1
vote
0answers
36 views

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 ...
0
votes
2answers
18 views

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 ...
1
vote
2answers
65 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 ...
0
votes
2answers
25 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 ...
5
votes
1answer
57 views

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 ...
0
votes
3answers
98 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 ...
3
votes
2answers
123 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 ...
1
vote
0answers
35 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 $A, AB, AC$.
1
vote
1answer
40 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 ...
1
vote
2answers
63 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 ...
0
votes
1answer
67 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 ...
0
votes
1answer
31 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 ...
3
votes
0answers
42 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 ...
3
votes
2answers
95 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 ...
0
votes
0answers
18 views

doubt in proving Ratio.

Find the equation of the tangent of the curve $x^4+y^4=a^4$ at $(h,k)$. And prove that the point of contact of divides the line segment joint the intercepts of the tangent in the ratio $h^3:k^3$. ...
2
votes
2answers
140 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 ...
1
vote
2answers
25 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 ...
0
votes
0answers
20 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 ...
1
vote
2answers
29 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: ...
1
vote
2answers
50 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 ...
0
votes
1answer
20 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 ...
1
vote
0answers
42 views

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 ...
0
votes
0answers
38 views

Books on vector analytic geometry

I'm looking for books about analytic geometry which covers affine change of coordinates,equivalence of conics by affine and projective change of coordinates, etc. using vectors. I would like a book in ...
1
vote
1answer
77 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 ...
0
votes
1answer
20 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, $$\arctan \dfrac{2\sqrt{h^2-ab}}{a+b}$$ ...
1
vote
1answer
105 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 ...
0
votes
0answers
35 views

I'm having troubles to find this parametrization.

I'm reading the Reid's Undergraduate Algebraic Geometry book of algebraic geometry for undergraduates and I have two questions about a proof of an example on the page 19: Red question: Reid said ...
0
votes
0answers
10 views

Prove that the intersections of the ray $f(x)\rightarrow x$ with the $n$-disk form a continuous function, with $f$ continuous

This appeared in a proof of the Brouwer fixed point theorem, in Introduction to Algebraic Topology, by Rotman, but it was left as an exercise. I could only prove this in 2 and 3 dimensions, not in ...
0
votes
1answer
23 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?
1
vote
0answers
18 views

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 ...
1
vote
2answers
17 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, ...
1
vote
1answer
28 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 ...
0
votes
0answers
16 views

Geometry involving area of rhombus and interior isosceles triangles

Points E, F, G, and H lie inside a rhombus ABCD, such that the triangles AEB, BHC, CGD, and DFA are isosceles right triangles with hypotenuses AB, BC, CD, and DA.The sum of areas of ABCD and EFGH is ...
1
vote
1answer
21 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 ...
3
votes
0answers
33 views

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, ...
0
votes
0answers
24 views

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 | ...
0
votes
0answers
21 views

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 ...
0
votes
1answer
24 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 ...