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

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3
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1answer
56 views

Segments on a plane, what curve do the intersections tend to?

In a Cartesian diagram, given a size $s$, suppose I create $m$ segments as such: I connect $(0,s/m)$ with $(s,0)$; $(0,2s/m)$ with $(s-s/m,0)$; ... ; $(0,s)$ with $(s/m,0)$. For example, if $s=4$ ...
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0answers
522 views

Calculation of the coordinates on the surface of a tilted cone

I have a mathematical problem (which I am trying to solve with Mathematica). I want to tilt a cone around its base point as in my example, where I have used Mathematica's Cone-function and spherical ...
0
votes
1answer
18 views

What is $a$ in the formula for the distance of point to plane formula: $h=\frac{|(a-p) \cdot n|}{|n|}$?

What does $a$ stand for in the following formula for the distance of a point to a plane? $$h = |PF| = \frac{|d - p \cdot n|}{|n|} = \frac{|(a-p)\cdot n|}{|n|} .$$
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2answers
197 views

Hyperbola property

I am posting the following question under homework category. I hope I will have very good answer from mathematicians about conic sections. I have seen closely the conic sections and their ...
1
vote
1answer
222 views

This is the most difficult question I could get without using mass point geometry

In triangle ABC, points D and E are on sides BC and CA respectively, and points F and G are on side AB with G between F and B. BE intersects CF at point O_1 and BE intersects DG at point O_2. If FG ...
3
votes
3answers
302 views

Apostol Section 13.25 #13 - Conic Sections

Question: Prove that a similarity transformation (replacing $x$ by $tx$ and $y$ by $ty$) carries an ellipse with center at the origin into another ellipse with the same eccentricity. (The next ...
-2
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1answer
1k views

How to prove we could use mass point geometry to solve all the triangle problem involving ratio between line segment and transversal in a triangle?

what is an easy way to prove that use mass point geometry to solve a problem in the link i provide that is involving cevians in a triangle is same as using the other way in euclidean geometry or ...
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2answers
39 views

Curve equation - help in understanding

OK, I wasn't on a class regarding this type of excercises. I got the notes from the lesson but have no idea how is it working. I hope you'll be able to clarify: Determine the equation of the curve ...
1
vote
2answers
277 views

Is there a generalized method of rotation for curves?

I know that we can rotate a curve in $R^2$ about a linear axis, as is common for first year calculus problems involving solids of revolution. But has anyone come up with a general method to take a ...
1
vote
0answers
369 views

Problem solving a set of quadratic equations

Sorry for being a newbie barging in with a question, but I'm facing a rather trivial problem which I seem unable to solve... Not being a matemathician (but an engineer with a bit of knack for math), I ...
1
vote
1answer
65 views

If $\|\mathbf{OA}+k\mathbf{OB} \|=1$, prove that $\text{Area}(OACB) \leq \| \mathbf{OB} \|$

OACB is a parallelogram. In other words if $\left \|\mathbf{a}+k\mathbf{b} \right \|=1$ ($k\in\mathbb{R}$), prove that $$\|\mathbf{a}\| \cdot \|\mathbf{b} \| \cdot \sin \theta \leq \|\mathbf{b} \| $$ ...
0
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1answer
471 views

How to solve such an equation ? (Line-Plane Intersection)

I don't know how to solve such an equation: $$ t = - \frac{ ...
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6answers
1k views

Why, conceptually, do limaçons $r=a+b\cos\theta$ have dimples when $|\frac{a}{b}|<2$?

Using calculus, I can justify that limaçons—the polar graphs of $r=a+b\cos\theta$ for various nonzero real values of $a$ and $b$—are dimpled when $|\frac{a}{b}|<2$, but that doesn't seem to yield ...
4
votes
2answers
349 views

Why do definitions of distinct conic sections produce a single equation?

I understand how to get from the definitions of a hyperbola — as the set of all points on a plane such that the absolute value of the difference between the distances to two foci at $(-c,0)$ and ...
0
votes
1answer
384 views

Line-line intersection derivation

I wanted to derive the formula to give the point of intersection of two lines, each defined by a pair of points. I got the wrong answer and cannot find the error. Which drives me crazy. I don't how ...
2
votes
2answers
465 views

Analytical calculation of the resulting surface between two overlapping spherical caps

Let's say I have a sphere (determined by its center and radius) and two planes which cut individually the sphere. Individually, there will be to spherical caps. Let's suppose that both spherical caps ...
6
votes
3answers
1k views

Parametric form of an ellipse given by $ax^2 + by^2 + cxy = d$

If $c = 0$, the parametric form is obviously $x = \sqrt{\frac{d}{a}} \cos(t), y = \sqrt{\frac{d}{b}} \sin(t)$. When $c \neq 0$ the sine and cosine should be phase shifted from each other. How do I ...
2
votes
4answers
1k views

Parametric equation for a plane perpendicular to a vector

The implicit equation for a plane perpendicular to a given vector at the origin is $ax + by + cz = 0$. I can write this in parametric form as $x = t, y = u, z = -\frac{at + bu}{c}$. The only problem ...
2
votes
4answers
1k views

Proving two lines trisects a line

A question from my vector calculus assignment. Geometry, anything visual, is by far my weakest area. I've been literally staring at this question for hours in frustrations and I give up (and I do mean ...
3
votes
3answers
156 views

Determine a point

$$\text{ABC- triangle:} A(4,2); B(-2,1);C(3,-2)$$ Find a D point so this equality is true: $$5\vec{AD}=2\vec{AB}-3\vec{AC}$$
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votes
5answers
1k views

Find extra arbitrary two points for a plane, given the normal and a point that lies on the plane

For a plane, I have the normal $n$, and also a point $P$ that lies on the plane. Now, how am I going to find extra arbitrary two points ($P_1$ and $P_2$) for the plane so that these three points $P$, ...
6
votes
3answers
8k views

Orthogonal projection of a point onto a line

please give me a directions how to solve this: find an orthogonal projection of a point T$(-4,5)$ onto a line $\frac{x}{3}+\frac{y}{-5}=1$
3
votes
5answers
586 views

Help understanding cross-product

I am trying to calculate the intersection point (if any) of two line segments for a 2D computer game. I am trying to use this method, but I want to make sure I understand what is going on as I do it. ...
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votes
4answers
1k views

Best way to find the Coordinates of a Point on a Line-Segment a specified Distance Away from another Point

I have 4 points: $Q, R, S, T$. I know the following Coordinates for $R$, $T$, and $S$; Length of $\overline{RQ}$ That segment $\overline{RT} < \overline{RQ} < \overline{RS}$; I need to ...
0
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1answer
393 views

algebraic way to compute intersection of disks

Is there a pure algebraic way to calculate intersection of two disks (extended to spheres, ellipses)?
2
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1answer
347 views

Distance between line and point in vector form

I am not asking anyone to do this for me. This question pops out of the blue, the ones before and after are trivial in comparison. I need hints: If $\vec{p}$ is a fixed point and $\vec{x}(t) = ...
5
votes
1answer
1k views

Formula for curve parallel to a parabola

I have a simple parabola in the form $y = a + bx^2$. I would like to find the formula for a curve which is parallel to this curve by distance $c$. By parallel I mean that there is an equal distance ...
3
votes
3answers
1k views

Canonical to Parametric, Ellipse Equation

I've done some algebra tricks in this derivation and I'm not sure if it's okay to do those things. $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = \cos^2\theta + ...
3
votes
3answers
193 views

How to calculate the x/y coordinate of F in this diagram (geometry)

In the diagram I've provided, how do I calculate the x,y coordinate of F if the points A,B,C are arbitrary points on a grid? I'm looking for a formula to solve F's X axis and another formula to ...
3
votes
1answer
157 views

Simulation of bouncing circles

I want to simulate two circles bouncing off one another. For this I am not sure what I need to calculate. I couldn't find any useful information on the internet, so I have thought long and hard about ...
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votes
6answers
1k views

Where can I find Linear Algebra in Nature?

I'm a Computer Science major and I've been studying Analytic Geometry and Linear Algebra this semester. Today my teacher gave a hell of an explanation talking about linear systems, quadratic ...
2
votes
1answer
195 views

Easier way to calculate this point besides line intersection?

Given are all points except E, plus |AF| = |DC|. Considering that the lines AB and FE, as well as BC and ED are parallel, is there an easier way to calculate E? Maybe some relation with B? I'd ...
4
votes
1answer
2k views

Analogue of spherical coordinates in $n$-dimensions

What's the analogue to spherical coordinates in $n$-dimensions? For example, for $n=2$ the analogue are polar coordinates $r,\theta$, which are related to the Cartesian coordinates $x_1,x_2$ by ...
2
votes
1answer
261 views

find increment amount to get from $(x_1,y_1)$ to$(x_2,y_2)$ one dot at a time

If I have two points on positive cartesian coordinates, how do I find: The slope of a line between those points the increment amount to get from $(x_1,y_1)$ to$(x_2,y_2)$ one dot at a time. What I ...
1
vote
1answer
270 views

Are these sufficient conditions to define an elliptical cone?

I was successful in deriving the equation for an elliptical double-napped cone in rectangular coordinates. All I did was define a line with slope $a$ on the xy-plane, and another line of slope $b$ on ...
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3answers
522 views

Distance between skew lines - correct method ?

If we have two skew lines in $\mathbb R^3$, $\vec r_{1} = \vec a + \lambda\vec d_1$ and $\vec r_{2} = \vec b + \mu\vec d_2$ then at their closest point, the difference vector $\vec r_2 - \vec r_1$ is ...
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1answer
281 views

questions from vectors applications

suppose that An boat captain wants to travel due south at 40 knots. If the current is moving northwest at 16 knots, in what direction and magnitude should he work the engine? here is given picture ...
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1answer
575 views

rotational of polynomials

first of all it is well known that if we rotate (x,y) coordinate by some angle (let's say by A) then new image(x',y') will be related to (x,y) by the following formula ...
5
votes
3answers
2k views

How is the angle between 2 vectors in more than 3 dimensions defined?

I would like to know how the angle between two n-vectors is defined. I mean whether it is unique and how we may compute it (is the inner product a valid method in the n-dimensional space?). I have ...
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2answers
2k views

rotation by 180 angle

in general i know that if we rotate (x,y) about origin by the 180 degree we will get new image (-x,-y),but suppose that we make rotation not about origin but some other point (a,b) does your result ...
2
votes
1answer
100 views

Finding the location of the end of an arc, knowing the beginning, the arc's length and the radius

I apologise in advance if this is really basic. I have a circle of radius 15, from which i work out an arc, given an angle of arbitrary value (it's for a computer program). Given that i know the point ...
2
votes
2answers
1k views

Finding the intersection of a two points and an arbitrary axis

Given two points I would like to find where the line joining them intersects an arbitrary axis. For example, if I had one point (5, 10) and another at (50, 100) I can be sure that somewhere a line ...
2
votes
2answers
4k views

How do we prove the rotation matrix in two dimensions not by casework?

I was trying to prove: To carry out a rotation using matrices the point $(x, y)$ to be rotated from the angle, $θ$, where $(x′, y′)$ are the co-ordinates of the point after rotation, and the formulae ...
4
votes
3answers
199 views

What is the most direct way to derive an equation for a parabola from its x and y intercepts?

I have a pair of points at my disposal. One of these points represents the parabola's maximum y-value, which always occurs at x=0. I also have a point which represents the parabola's x-intercept(s). ...
4
votes
5answers
12k views

Determining if an arbitrary point lies inside a triangle defined by three points?

Is there an algorithm available to determine if a point P lies inside a triangle ABC defined as three points A, B, and C? (The three line segments of the triangle can be determined as well as the ...
5
votes
2answers
252 views

Locus and concurrent lines

This will be my first question :-) Let $\mathcal{D}_1$ and $\mathcal{D}_2$ two concurrent lines, and $F$ a point in the plane, and $H$ and $G$ its images by the symmetries of axis $\mathcal{D}_1$ and ...
3
votes
1answer
239 views

Recomputing arc center

Please excuse my poorly drawn doodle here, I'm almost inept at drawing. I'm attempting to compute i2, j2, x2, y2. Knowns: x1, y1, xk, yk, i1, j1, the arc is circular Constraints: resulting arc ...
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vote
2answers
197 views

Distribution or bounds for maximum Cartesian coordinate sampled from the sufarce of an n-sphere

It's been said that for high dimensions a hypersphere is "nearly all equator". The amount of space near the poles is just ridiculously small. This of course means that from a uniformly random sample ...
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2answers
907 views

Analytic Geometry: Point coordinates, same distance from two points

Given are two points, $P1(x_1, y_2)$ and $P2(x_2, y_2)$, and distance $a$. Now I want to find the two points $T1$ and $T2$. $$d(P1,T1) = d(P2, T1) = a = d(P1,T2) = d(P2, T2)$$ Eg: (T1 and T2 are my ...
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2answers
175 views

Finding a random vector exactly yay far from another point in 3D space

So I am trying to find a vector a certain distance away from another point ( the distance varies based on an input ) and I've figured out that ...