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

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5
votes
1answer
194 views

Cross Product Intuition

I know the cross product between a vector $a$ and a vector $b$ is just a vector whose magnitude is the product of magnitude of $b$ times the magnitude of the perpendicular component of $a$ in relation ...
3
votes
1answer
859 views

Intersection between a plane and a sphere

We have a sphere $(x-1)^2 + (y-1)^2 + (z-1)^2 = 1$ and a point $A = (1;1;-1)$. Find all equations of planes which contain the line $OA$ and intersections with the sphere are circles of radius ...
2
votes
2answers
182 views

Circles of radius $2$ passing through origin with centers on $x=1$

There are two circles of radius $2$ that have centers on the line $x=1$ and pass through the origin. Find their equations. Please explain to me what the problem is really saying.
0
votes
2answers
430 views

Check my answers to the problems related to analytic-geometry

1) Find the equation of the circle of radius $2$ with center at $(3, 0)$. My answer: $\sqrt{(x-3)^2 + y^2} = 2$ 2) Find the equation of the circle of radius $\sqrt3$ with center at (-1, -2). ...
0
votes
1answer
60 views

Problem of sketching a circle

I've to solve a problem in which I've been given this equation: $x^2 + y^2 = 4$ and I've to sketch a circle which is the locus of the equation. Here $2$ is the radius $r$ of the circle. $2$ doesn't ...
2
votes
2answers
2k views

Detecting an Intersection between Simple Shapes

I have a circle, ellipse, square or a rectangle, and I want to determine if it intersects a given triangle. I am looking for the easiest way to determine if there exists a geometrical intersection ...
2
votes
3answers
1k views

Locus of the equation

One way to describe a set of points in the plane is by an equation or inequality in two variables, say $x$ and $y$. A solution of an equation in $x$ and $y$ is point $(x_0, y_0)$ in the plane for ...
0
votes
3answers
593 views

Distance between two points

The distance between the two points $P(x_1, y_1)$ and $Q(x_2, y_2)$ is the quantity $$\mathrm{distance}(P, Q) = \sqrt{(\Delta x)^2 + (\Delta y)^2}.$$ Is $(P, Q)$ above indicating an open ...
3
votes
2answers
71 views

Difference of two points on a plane

If $P(x_1, y_1)$ and $Q(x_2, y_2)$ are the two points on a plane, then the change in $x$ and $y$ coordinates is denoted by $∆x$ and $∆y$ respectively. Therefore, $x = ∆x = x_2 - x_1$ and $y = ...
3
votes
1answer
145 views

Confusion between an ordered pair, and open interval

The $(x, y)$ plane is the set of all ordered pairs $(x, y)$ of real numbers. The origin is the point $(0, 0)$. The $x$-axis is the set of all points of the form $(x, 0)$, and the $y$-axis is ...
3
votes
1answer
2k views

How is the formula for the focal point of a ball lens derived?

How can the focal point of a ball lens be found?
0
votes
0answers
49 views

Geometric calculation: two kneading discs

I have two kneading discs of a screw overlapping with each other at 60 deg. I know the cross section area of one disc and I want to know what will be the overlap area if the other disc is rotated at ...
3
votes
1answer
317 views

$2D$ Line Segment - Triangle Intersection

I've seen similar questions but could not solve my problem with those. My question is how to detect an intersection of a line segment and a triangle on a 2D coordinate system? I don't need the point ...
2
votes
2answers
328 views

Algebraic solution to find circle radius given distance of three external points from perimeter

I have an engineering problem, which involves math. The reason it's "engineering" is that I don't need a pure mathematical solution, but a good-enough approximation could work - the only constraint is ...
0
votes
1answer
75 views

Equation of a plane passing through a line and a separate point.

Is it possible to find the equation of a plane that passes through a line and a point not on the line? For example, the line $y=7x-7$ and the point $x=3,y=0,z=8$. I've tagged this as homework, but ...
4
votes
1answer
46 views

How does this method to find the centre work?

Say we have a conic with equation $f(x,y)=c$. My teacher says that it's centre satisfies the equations : $f_x(x,y)=f_y(x,y)=0$ (If it has a centre). She didn't give any explanation. I thought this ...
0
votes
3answers
85 views

Prove that $|PC|^2 + |PD|^2 = |AB|^2$ if

We have an angle of 90° so that there are 2 points A, B on each side of the angle, O is the vertex and |OA| = |OB|. On the arc AB with it's center being in O, we pick an arbitrary point P and draw a ...
0
votes
1answer
297 views

Find a locus of points to satisfy these conditions?

So, we have to straight lines: $$3x-4y+5=0$$ $$2x+3y-4=0$$ You have to find a locus of points from which all perpendiculars to the two lines given are in a 2:3 ratio.
2
votes
2answers
238 views

proving or disproving that two tangent lines are parallel to a curve

Im trying to prove or disprove that given the function, $f(x)=0.5\sqrt{1-x^{2}}$, There are two different tangent lines to $f(x)$ so they are parallel. I tried to derivative but with no success.
5
votes
2answers
94 views

If $\left |z-3\right |=\left |z+i\right |$, where $z=x+iy$, prove that $3x+y=4$

If $\left |z-3 \right |=\left |z+i\right |$, where $z=x+iy$, prove that $3x+y=4$. I have got to the point where I have $\left |z \right |= \sqrt{x^2+(y+1)^2} = \sqrt{(x-3)^2+y^2}$ But really ...
2
votes
2answers
56 views

Compute the value of the exterior $2$-form

Compute the value of the exterior $2$-form $$\omega = (x_1 + x_2)e_1^* \wedge e_2^* + (x_2 + x_3)e_2^* \wedge e_3^* + \cdots + (x_i, x_{i+1})e_i^* \wedge ...
1
vote
0answers
118 views

Parametrizing a section of a torus

Consider the torus obtained by rotating the circle $(x-R)^2+z^2=r^2$ about the $z$-axis, where $R>r>0$. Parametrize the part of this torus where $z>x+y$. My approach to this so far is to ...
1
vote
2answers
318 views

Max and min value of $7x+8y$ in a given half-plane limited by straight lines?

So, there are four inequalities: $$\begin{eqnarray*} y &\geq &-3x+15; \\ y &\leq &-11/3x+56/3; \\ x &\geq &0; \\ y &\geq &0. \end{eqnarray*}$$ If we draw all those ...
1
vote
0answers
37 views

Is there a way to check if my Taylor Expansion is correct?

I have an exam later and I need to do Taylor expansions of functions. I have questions like: Consider the map $F:\mathbb{R}^2_x \rightarrow \mathbb{R}^2_y$, given by the equations $$y_1 = ...
1
vote
4answers
256 views

A triangle has corners with coordinates (1,2), (-2,3) and (0,-1). Please help me determine the equations of the lines.

A triangle has corners with coordinates (1,2), (-2,3) and (0,-1). Please help me determine the equations of the lines that form the sides of the triangle.
0
votes
1answer
121 views

finding the locus of quotient lengths of tangents to circles

I'm trying to find the locus of all points so that their quotient of the Tangents lengths to circles: $x^2+y^2-12x=0, x^2+y^2+8x-3y=0$ is $2:3$, respectively. i tried to use the formula: ...
10
votes
2answers
115 views

Prove that $|PF_{1}|+|PF_{2}|$ is Constant in an Elipse

Given an elipse with two focus $F_{1}$ an $F_{2}$, and $A$ is an arbitrary point at the elipse. Stright line $AF_{1}$ has another intersection point $B$ with the elipse, and $AF_{2}$ has another ...
4
votes
1answer
194 views

The intuition behind the definition of geodesics on a Riemannian manifold. (A non-technical question)

In the text I'm studying, the idea behind the definition of a geodesic on a Riemannian manifold was sketched via paths in $\mathbb{R}^n$. I have trouble understanding some aspects of it. Let $\gamma: ...
3
votes
1answer
407 views

Draw an arc in 3d coordinate system

I have some legacy code which is supposed to draw an arc with constant radius in 3d space however it is drawing the arc in the wrong position. I would like to know and understand the mathematical ...
0
votes
0answers
113 views

two points on a unit sphere

Consider the two vectors to the points on the unit sphere, $${\bf v}_i=(\sin\theta_i\cos\varphi_i,\sin\theta_i\sin\varphi_i,\cos\theta_i)$$ with $i=1,2$. Use the dot product to get the angle $\psi$ ...
1
vote
1answer
276 views

shortest distance between two points on $S^2$

Length of Curve in $2D$ is $l_{\gamma}(\mathbb{R}^2)=\int_{0}^{1}\sqrt{(dr/dt)^2+r^2(d\theta/dt)^2}$ Length of a curve in $3D$ is ...
4
votes
2answers
628 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
501 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
146 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
320 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
50 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
110 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
270 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
28 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
223 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
509 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
122 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
107 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
91 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
679 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
238 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
506 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
236 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
659 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
102 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 ...