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

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3
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2answers
9k views

How do I find the equation of a tangent line to a curve?

I'm given $x^2+2x-4$ at $x=2$ and I have to find the tangent line to this curve at that point...
0
votes
1answer
132 views

lines intersection

I have to find intersection of two lines ($AH$ & $CD$) $A(3.42,-1.84,8.56) $ $B(-3.42,3.84,-8.56) $ $C(0.00,16.25,0.00)$ $AH$ is the perpendicular; $CD$ is the median I tried so: Firstly I ...
-1
votes
1answer
141 views

Formula to show square root between 2 values

Bear with me - it's been a while since I did this at school! I need to plot a curve in the form of a square root (kind of an 'r' shape if you will) I have 6 intervals along my x axis, and my maximum ...
1
vote
1answer
105 views

Mirorring a set of Points

Let's say I have a cloud of points, and I know the equation of the symmetry plane. I'd like to mirror every single point with respect to this plane. It might be much simpler than I think, but I have ...
2
votes
1answer
365 views

Need help with the proof of conic section

Prove that the intersection of a plane and a object consist of one cone and one upside-down cone where the tip of cone meet is either degenerate conic or conic Also, idenify in what situation, the ...
2
votes
1answer
198 views

What is wrong with this proof that isometries must be surjective?

Let $\phi : \mathbb{R}^2\rightarrow\mathbb{R}^2$ be an isometry. Suppose $\phi$ is not surjective, that is there exists some $v \in \mathbb{R}^2$ whose fiber $\phi^{-1}(v)$ is empty. Then by the ...
0
votes
2answers
141 views

Solving this kind of equation

Say you have two equations with three variables, the first is the equation of the surface of a sphere and the second of a plane. In this case they intersect in a point $(1,0,0)$. The only way I know ...
2
votes
0answers
465 views

projection of a sphere onto a plane

Consider you have a sphere centered at the origin.The sphere has a diameter of $\frac{1}{2} \sqrt{\frac{3}{2}}$. This means that the inscribed cube has an edge of 1. Take any point from the plane ...
5
votes
5answers
1k views

Distance Between A Point And A Line

Any Hint on proving that the distance between the point $(x_{1},y_{1})$ and the line $Ax + By + C = 0$ is , $$\text{Distance} = \frac{\left | Ax_{1} + By_{1} + C\right |}{\sqrt{A^2 + B^2} }$$ What ...
4
votes
3answers
195 views

Points at integer distance

How many points can one can place in $\mathbb{R}^n$, with the requirement that no $n+1$ points lie in the same $\mathbb{R}^{n-1}$-plane, and the euclidean distance between every two points is an ...
0
votes
2answers
103 views

Calculate Points for a Parallel Line

Given a line running through p1:(x1,y1) and p2:(x2,y2), I need to calculate two points such that a new parallel line 20 pixels away from the given line runs through the two new points. Edit: The ...
0
votes
1answer
959 views

Find point on sphere with directional tangent vector

Say a sphere equation like this: $x^2+y^2+z^2=5$. I want to find a point on the sphere whose tangent vector is perpendicular to the vector $\begin{bmatrix} 2\\ 3\\ 4 \end{bmatrix}$. I go ...
3
votes
1answer
55 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 $s=4$ ...
1
vote
0answers
513 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? $$h = |PF| = \frac{|d - p \cdot n|}{|n|} = \frac{|(a-p)\cdot n|}{|n|} .$$
0
votes
2answers
190 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
300 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
votes
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 ...
1
vote
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
270 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
365 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
votes
1answer
461 views

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

I don't know how to solve such an equation: $$ t = - \frac{ ...
5
votes
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 ...
3
votes
2answers
343 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
377 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
453 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
972 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
155 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: $$5\vec{AD}=2\vec{AB}-3\vec{AC}$$
5
votes
5answers
2k 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
9k 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
579 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. ...
2
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
votes
1answer
391 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
votes
1answer
346 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 ...
6
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
194 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
3k 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
256 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
268 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 ...
1
vote
3answers
510 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 ...
1
vote
1answer
270 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 ...
0
votes
1answer
540 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 ...