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

learn more… | top users | synonyms (1)

4
votes
2answers
600 views

Length of curve in 3D spherical coordinate

let $r$ be the magnitude of a vector in 3D with Spherical co-ordinate $(r,\theta,\phi)$ and cartesian coordinates is $(x,y,z)$, whose angle with $z$ axis is $\phi$ and projection of the vector makes ...
1
vote
2answers
476 views

How can I calculate the Euclidian displacement of two places on a sphere (earth in this case ) and calculate the

I would like to get the formula on how to calculate the distance between two geographical co-ordinates on earth and heading angle relative to True North. Say from New York to New Dehli , I draw a ...
-1
votes
1answer
144 views

Length of a curve on $S^2$

$1.$ Could any one tell me what is the shortest distance between $2$ points on $S^2$? $2.$ Could any one tell me how to measure explicitly a length of a curve on the $S^2$ using polar co-ordinates? ...
3
votes
1answer
316 views

Common area of two rectangles

Suppose we have a rectangle at the center of the coordinates. One top point of the rectangle has the coordinates (a, b), the second (-a, b), third (-a, -b) and (a, -b). We rotate this rectangle with ...
0
votes
2answers
49 views

Points in common

I have the following problem: How many points do the graphs of $4x^2-9y^2=36$ and $x^2-2x+y^2=15$ have in common? I know that the answer is in the system of two equations, but how should I solve it? ...
1
vote
3answers
109 views

Proof: Two non identical circles have at most 2 same points

I'm struggeling with an analytic proof for the fact, that two different circles have at most 2 same points. (I try to solve it analytical, because geometrical I already prooved it). I tried to start ...
1
vote
3answers
265 views

Intersection points of a Triangle and a Circle

How can I find all intersection points of the following circle and triangle? Triangle $$A:=\begin{pmatrix}22\\-1.5\\1 \end{pmatrix} B:=\begin{pmatrix}27\\-2.25\\4 \end{pmatrix} ...
0
votes
1answer
27 views

Paramertrization of intersection between spehere and plane.

I have the normal $n = (a,b,c)$ for a plane through origo,and want to find the paramertrization of the unit circle. How can I do this? I guess I should eliminate one coordinate from the plane and ...
3
votes
1answer
217 views

How to find the smallest enclosing ellipse around two circles?

Given two circles (defined by center and radius), how do I find the smallest ellipse that encloses both of them? I.e. I search the green ellipse in the picture below. The ellipses can be considered ...
5
votes
1answer
497 views

What are some isometries of $S^2$ without fixed points?

This spherical geometry question involves isometries. I am particularly looking for isometries with no fixed points.
3
votes
1answer
119 views

Complex Numbers - Locus

Suppose that $k|z-z_1|=l|z-z_2|$ where $k\neq l$ and both are positive real numbers. Show that the locus of $z$ in the Argand diagram is a circle with center: $$\frac{k^2 z_1-l^2 z_2}{k^2-l^2}$$ and ...
0
votes
1answer
105 views

How to get Euler angles where an initial value of Euler angle is set as baseline

I have a sensor which gives me Euler angles (roll,pitch,yaw). There is a baseline value of Euler angle (assume it is 5,10,15) at the beginning.I want to calibrate this baseline values from all ...
0
votes
1answer
90 views

$\tan{\frac{a}{2}}\cdot \tan{\frac{b}{2}}\cdot \tan{\frac{c}{2}}\leq \frac{1}{3\sqrt{3}}$, Where a,b,c are angles of triangle

As in title $$\tan{\frac{a}{2}}\cdot \tan{\frac{b}{2}}\cdot \tan{\frac{c}{2}}\leq \frac{1}{3\sqrt{3}}$$whats more, is that this is acute triangle. I think it should be doable somehow with Jensen ...
2
votes
1answer
646 views

The definition of distance and how to prove the ruler postulate in Euclidean geometry

I have started to read some books about geometry. At the moment I've just started to read Hilbert's axioms and also some elementary books for highschool. From the basic perspective of an axiomatic ...
2
votes
1answer
231 views

Foundations of analytic geometry

I was just wondering about the formal foundations of analytic geometry, I mean axiomatically. I've noticed along my course of linear algebra that the axioms of vectorial space already include the fact ...
0
votes
2answers
496 views

Calculate Spherical Distance between points

I have googled this and not come up with an answer yet, but basically, I'm trying to find out the distance between each point or vertice on a sphere (all points are evenly spaced). I already know this ...
0
votes
2answers
224 views

Non-degenerate quadratic form and non-singular matrix

Let $(V,Q)$ be a finite-dimensional quadratic space over a field $\mathbb{K}$. From my definition, $Q$ is non-degenerate if $\operatorname{rad}(V)=\{0\}$. How can I prove that $Q$ is non-degenerate ...
3
votes
0answers
646 views

Turning radius of a vehicle

What's the minimum turning radius of a vehicle, rectangular in shape, with length l units and width w units? One key point to consider, would be that, the inclination of the front wheels can be ...
-1
votes
2answers
100 views

Is it possible to find the coordinates of a point in 3D space, given its distance from a known point?

Is it possible to find the coordinates $(x,y,z)$ of a point in $3d$ space when given: A) the unknown point is $(x,y,z)$. B) the known point is $(a,b,c)$. C) the distance between the two points is ...
1
vote
2answers
60 views

Determining a point's coordinates on a circle

So I have a circle (I know its center's coordinates and radius) and a point on the circle (I know its coordinates) and I have to determine the coordinates of another point on the circle which is ...
0
votes
0answers
21 views

Divide line in $XY$-dimension

For example we have line $A$ with coordinates $(0, 1, 10, 9)$; And we need to divide this line by $3$ (so we have now $A_1, A_2$ and $A_3$), where $A_1 + A_2 + A_3 = A$; Is there equation, to find ...
1
vote
1answer
144 views

Vector Function Magnitude

I was wondering, when you take the magnitude of the vector function $r(t)$, what does it represent geometrically? Does it represent the magnitude of the displacement vector, whose initial point is ...
0
votes
1answer
377 views

Analytical geometry - circles

How do you find the point for a circle and find the radiums when x squared has a co-efficient?
2
votes
1answer
101 views

Prove that sum of 2010 vectors is $\neq 0$ if these vector create a set with lengths numbers $\{1,2,\ldots,2010\}$

A set $V$ has 2010-vectors: $V=\{v_{1}, \ldots,v_{2010}\}$ and these vectors create another set with the lengths of these vectors: $B=\{1,2,\ldots,2010\}$. Each vector is parallel to one of $2$ given ...
0
votes
1answer
587 views

find the area of a parallelogram with the sides are given using the fourth standard equation of straight line

the sides of a parallelogram are on the lines $$x-3y+20=0,\\ x+y+6=0,\\ x-3y-10=0 \text{ and} \\ x+y+2=0.$$ Find its area. solve using the fourth standard equation of the straight line.
0
votes
1answer
315 views

Max. distance of Normal to ellipse from origin

How Can I calculate Maximum Distance of Center of the ellipse $\displaystyle \frac{x^2}{a^2}+\frac{y^2}{b^2} = 1$ from the Normal. My Try :: Let $P(a\cos \theta,b\sin \theta)$ be any point on the ...
4
votes
1answer
311 views

How to calculate the area closed by a parabola and a line without calculus?

In order to simplify the problem, suppose we have a parabola $y=ax^2+bx+c$, here $a\neq0$, and a line $y=kx+d$, here $k\neq0$. We can assume that they will intersect at two different points. Thus, the ...
1
vote
1answer
48 views

Unit vectors orthogonal to L

I have a line $L$ in $\mathbb{R}^2$ that passes through two points: $u = [9;7]$ $v = [1;-5]$ How do I find all unit vectors orthogonal to $L$? I know: $[x;y] * [8;12] = 0$ and $x^2 + y^2 = 1$ ...
2
votes
3answers
1k views

Find all unit vectors orthogonal to line with two given points

I have a line $L$ in $\mathbb{R}^2$ that passes through two points: $[9;7]$ and $[1;-5]$ How do I find all unit vectors orthogonal to $L$?
16
votes
1answer
250 views

Q: Given the graph of $y = \frac{1}{x}$, construct the $(x,y)$ coordinate axes using straightedge and compass

The solution to the problem above is known (see comments for a hint). What other analytic functions can one substitute for $y = \frac{1}{x}$, and still be able to do so?
0
votes
1answer
38 views

A Statement About Points in the Real Euclidean Space

Suppose that $n \geq 3$, $x$, $y \in \mathbf{R}^n$, $d \colon= |x-y| > 0$, and $r>0$. Then how to prove the following assertions: (a) If $2r>d$, there are infinitely many $z \in ...
0
votes
3answers
607 views

Find the length of this chord.

I've been trying to solve this geometry question for past 2 hours but haven't got the answer yet. There are two concentric circles or radius $8 cm $ and $13 cm$ with the common center $O$. $PQ$ is ...
1
vote
0answers
156 views

How to generalise a result regarding intersections of cones and other convex sets?

To test for a particular property of positive LTI systems using feasibility problems I've come across the following claim which, intuitively, I believe can be generalised. I think I've (rather ...
-1
votes
1answer
231 views

Show that the equilateral triangle has congruent angles?

This Question is of Chapter "Straight Line" the diagram of this question shows the values of ABC I am confused abut the values of C(x,y) it should be (b,c) but it's written something else can someone ...
0
votes
1answer
223 views

In an equation that looks like the standard form of an ellipse, what must the constant on the RHS equal for exactly one solution?

I am working on a homework question: What must be the value(s) of $c$ for the following equation to have exactly 1 solution? The equation is of the standard form of the equation for an ellipse, ...
7
votes
4answers
154 views

closest point to on $y=1/x$ to a given point

I feel like I'm missing something basic - given a point $(a,b)$ how do I find the closest point to it on the curve $y=1/x$? I tried the direct approach of pluggin in $y=1/x$ into the distance formula ...
0
votes
2answers
112 views

Find minimum distance

I came across this problem in a maths exam. I solved this by taking that a light ray passes in such a way that it takes least path. But as this was a maths exam, i was wondering if it can be solved ...
2
votes
3answers
415 views

Simultaneous Equations and Vectors

The question I am currently working on is, "...find $a$ and $b$ such that $\vec{v} = a \vec{u} + b \vec{w}$, where $ \vec{u} = \langle 1,2 \rangle$, $\vec{w} = \langle 1,-1 \rangle$, and $\vec{v} = ...
0
votes
2answers
57 views

Finding the equation of a plane.

How do I find the equation of a plane given by the points (0,1,1), (1,0,1) and (1,1,0)? Graphing it, it's a triangle when you connect the points. Can I use this somehow?
0
votes
1answer
138 views

How can I find the intersection of a line vector and a plane?

Here is my vector: $(-3,1,-4)+r(4,0,1)$ And my plane: Created from the following vectors: $x: (3,0,1)+t(-1,1,2)$ $x: (0,2,-1)+s(2,-2,-4)$ $(3,0,1)+t(-1,1,2)+n(2,-2,-4)$ (Cartesian: ...
8
votes
1answer
199 views

Maps of $\mathbb{R}^3$ preserving the cross product

Given a map $\phi:\Bbb R^3 \rightarrow \Bbb R^3$ such that for all $a,b \in \Bbb R^3$: $$\phi(a \times b)=\phi(a) \times \phi(b)$$ Is $\phi$ necessarily a rotation around the origin or the map ...
1
vote
2answers
90 views

How to find angle of plane $7x+13y+4z = 9$ with $xy$ coordinate plane?

How can I calculate inclination of $7x+13y+4z = 9$ with $X-Y$ plane As for as I understand from question is that the angle of plane $7x+13y+4z=9$ with $ax+by+0z=d$ for $(XY)$ plane.
0
votes
2answers
396 views

Q). Show that the four points are angular points of a rectangle$ (0,-1) (4,-3) (8,5) (4,7)$.

I started to solve the question by taking the sides of rectangle ABCD then added a midpoint in the rectangle and divided the rectangle in diagonal then found out the midpoint of diagonals AC and BD ...
2
votes
1answer
210 views

Tangent cone to a subset of $\mathbb{R}^3$

Well, I have the set $X=\{(x,y,z) \in \mathbb{R}^3 | 3x^2+2x^3+y^2+z^2=1\}$ How can I calculate the tangent cone at the point $(-1,0,0)$ ? What are the standard ways to calculate the tangent cone to ...
4
votes
0answers
89 views

Points at Integer Distances in 3-space

With the restriction no three points in a line, no four points on a circle, there is a 7 point configuration of points on the plane such that all pairs of points are at integer distances. [1] For ...
2
votes
2answers
178 views

Given an algebraic curve $F(x,y)=0$, why do the partial derivatives of $F(x,y)$ being zero at a point imply the plane curve has a singularity?

I'm looking at algebraic plane curves of the form $F(x,y)=0$ and trying to figure out why for points on the curve such that $\frac{\partial F}{\partial x} = \frac{\partial F}{\partial y}=0$, the plane ...
0
votes
2answers
31 views

Computation with scalar product

Let $\vec{a}$ and $\vec{b}$ be vectors from $V_3$. Suppose, that $|\vec{a}| = 1$, $|\vec{b}|=2$ and the angle between $\vec{a},\vec{b}$ is $\frac{\pi}{3}$. Use the properties of scalar product and ...
0
votes
1answer
101 views

Normal vector to surface

This is a very noob question, but can someone please give me an example of finding the normal vector to a surface (if this is the word in English) which is defined by three points in it. I know that ...
1
vote
3answers
2k views

how to find focal radius in parabola?

will we find focal radius in parabol, if our equation is $y^2=12x$. Do I need another variable? I have tried many times but I cannot find this problem. Thanks.
2
votes
2answers
84 views

Area of a decentered circunference [duplicate]

Possible Duplicate: Area of a portion of an arbitrarily-placed circle? Given a circunference of radius $R$ with the center in $P\equiv(x_0,y_0)$ $$(x-x_0)^2+(y-y_0)^2=R^2$$ I need to know ...