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

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0
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1answer
116 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
99 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
739 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
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1answer
267 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
533 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
288 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
746 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
114 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
65 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
23 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
162 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
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1answer
387 views

Analytical geometry - circles

How do you find the point for a circle and find the radius when $x^2$ has a co-efficient?
2
votes
1answer
103 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
751 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
390 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
497 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
51 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
253 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
40 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
754 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
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0answers
186 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
270 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
400 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
161 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
115 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
580 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
64 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?
13
votes
4answers
2k views

What is a point?

In geometry, what is a point? I have seen Euclid's definition and definitions in some text books. Nowhere have I found a complete notion. And then I made a definition out from everything that I know ...
0
votes
1answer
150 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
202 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
140 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
495 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
242 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
96 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
187 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
32 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
104 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
3k 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
85 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 ...
1
vote
1answer
258 views

Intersection of two lines

What is the suggested method to find the intersection of two line *segments in 3D space programmatically? I mean there are various methods to solve a set of 2 linear equations, eg. Using ...
2
votes
1answer
195 views

Metric tensor of complex numbers & Hamiltonian Mechanics

The Euclidean $\mathbb{R}^2$ geometric space can be mapped onto $\mathbb{C}$. In other words I see it like this $$\vec{v} = x\vec{x}+y\vec{y} = x\vec{1}+y\vec{i}= \begin{bmatrix}x \\y\end{bmatrix} ...
0
votes
1answer
144 views

At least two circles meeting these cond. have nonempty intersection

Here is a problem I've been trying to solve for some time now. Maybe you could help me. We have two sets $\mathcal {S}$ is a family of circles in the plane such that for any $x \in \mathbb{R}$ there ...
2
votes
1answer
63 views

Plugging in a point not on a plane, into the plane's equation

Say a plane P has a given equation ax+by+cz=d. Given a point $(x_0, y_0, z_0)$ that is not included in P. When $(x_0, y_0, z_0)$ is plugged into $f(x,y,z)=ax+by+cz-d$ and it outputs some nonzero ...
3
votes
2answers
602 views

Find the standard form of the conic section $x^2-3x+4xy+y^2+21y-15=0$

Find the standard form of the conic section $x^2-3x+4xy+y^2+21y-15=0$. I understand the approach in trying to solve these problems. But the $4xy$ is confusing me. I am not sure of where to start on ...
2
votes
0answers
116 views

Find $k$ such that the intersection of $x+ky=1$ and $y^2 - x^2 - z^2 = 1$ is an ellipse or a hyperbola

Find the values of $k$ such that the intersection of the plane $x+ky=1$ with the two-sheeted elliptic hyperboloid $y^2 - x^2 - z^2 = 1$ is (a) an ellipse and (b) a hyperbola. My attempt is the ...
2
votes
2answers
3k views

How can you construct as many intersections as possible with n lines?

If you have $n$ lines, it seems to be obvious that you can have at most $\frac{n^2-n}{2}$ intersections: $n = 1$: Obviously you need two lines to intersect, so the maximum number of intersections is ...
5
votes
4answers
10k views

Find intersection of two 3D lines

I have two lines $(5,5,4) (10,10,6)$ and $(5,5,5) (10,10,3)$ with same $x$, $y$ and difference in $z$ values. Please some body tell me how can I find the intersection of these lines. EDIT: By using ...
2
votes
1answer
381 views

Find a plane whose intersection line with a hyperboloid is a circle

Find a plane $\pi$ which involves x-axis and its intersection line with $$\frac{x^2}{4}+y^2-z^2=1$$ is a circle. Because the plane want to be find involves x-axis,so set as $By+Cz=0$,then I must to ...
1
vote
1answer
452 views

How to find the tangent cone of the sphere

A given sphere: $$x^2+y^2+z^2+2x-4y+4z-20=0$$ How to find the tangent cone of it ? the vertex of the cone is $(2,6,10)$ thanks very much.