Questions on the use of algebraic techniques to prove geometric theorems.
0
votes
1answer
20 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
95 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 ...
4
votes
1answer
149 views
what are some isometries of S^2 without fixed points?
Spherical geometry question involving isometries.
Particularly looking for isometries with no fixed points.
3
votes
1answer
82 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
63 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
68 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
65 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
63 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
287 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
63 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 ...
4
votes
0answers
400 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
vote
2answers
41 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
16 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
33 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 ...
-1
votes
2answers
59 views
three questions on analytic geometry and matrices
the lines $x-2y=4$ and $6x+ay=8$ are perpendicular. Calculate the value of $a$.
prove that the matrix $\pmatrix{\cos\theta& \sin\theta\\
-\sin\theta & \cos\theta}$ ...
0
votes
1answer
141 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
87 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
102 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
110 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
77 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
33 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
358 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 $L$?
7
votes
0answers
97 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
30 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
119 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
37 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 ...
0
votes
1answer
58 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
46 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
116 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
96 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
79 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
42 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
52 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: ...
7
votes
1answer
117 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 ...
2
votes
1answer
97 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
53 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
106 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
27 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
46 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
309 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
73 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
133 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
91 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
77 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
41 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 ...
0
votes
0answers
16 views
Feet of the three normals to the ellipsoid lie on a given plane how to find a plane in which other three normals will lie
Suppose normals are drawn from a point $(f,g,h)$ to the ellipsoid $(x/a)^2$+$(y/b)^2$+$(z/c)^2$=$1$
and if the feet of the three normals lie on the plane $(x/a)$+$(y/b)$+$(z/c)$=$1$ then how to ...
3
votes
2answers
204 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
43 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 ...
1
vote
2answers
116 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 ...
2
votes
4answers
250 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 ...
