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

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2
votes
1answer
139 views

Find the distance between two lines in $ \Bbb R^3 $ [duplicate]

There are two lines in $ \Bbb R^3 $ given in parametric form: $$ l_1: \left\{ \begin{aligned} x &= x_1 +a_1t\\ y &= y_1 +b_1t \\ z &= z_1 +c_1t \\ \end{aligned} \right. $$ $$ l_2: ...
1
vote
3answers
56 views

Convert a line in $ \Bbb R^3 $ given as intersection of two planes to parametric form.

We have a line in $ \Bbb R^3 $ given as intersetion of two planes: $$ \left\{ \begin{aligned} A_1x+B_1y+C_1z + D_1 &=0 \\ A_2x+B_2y+C_2z + D_2 &=0 \\ \end{aligned} \right. $$ How to ...
0
votes
1answer
551 views

find the locus of P(x,y)

Given point $A$ on the circle $x^2+y^2=R^2$. From $A$ passes parallel line to the x-axis. On this parallel line we Assign from point $A$ a Section with Length $2R$ at the Positive direction of the ...
5
votes
2answers
91 views

How does this vector addition work in geometry?

I saw the accepted answer to the question: Finding a point along a line a certain distance away from another point! I am not getting how to use it actually to find the coordinates of the new point at ...
1
vote
2answers
258 views

Parametrization of $x^2+y^2=z^2$

How can we show that any point on $x^2+y^2=z^2$ can be written in the form (z $\cos(\theta)$, z $\sin(\theta)$, z) for some $\theta$? Here is how I tried to approach it: $$(z \cos(\theta))^2+(z ...
11
votes
5answers
17k views

What is the general equation of the ellipse that is not in the origin and rotated by an angle?

I have the equation not in the center, i.e. $$\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1.$$ But what will be the equation once it is rotated?
5
votes
2answers
111 views

Why are close points in different regions close to the boundary?

I have a problem and it is bugging the heck out of me. It seems very obvious but now has become a major and frustrating stumbling block to a more productive line of thought. I have used a regular ...
3
votes
1answer
628 views

Enlarging an ellipses along normal direction

Given an ellipses, enlarge it along normal direction a fixed length say 1cm. Do we get another ellipses? If so, how to prove ?
1
vote
1answer
116 views

Rank of Jacobian at a singularity

Is the following proposition true? Proposition: Suppose $\mathbb{C}\{x_1,\ldots,x_{d_1}\}/(f_1,\ldots,f_{k_1}) \cong \mathbb{C}\{y_1,\ldots,y_{d_2}\}/(g_1,\ldots,g_{k_2})$ are isomorphic complex ...
1
vote
1answer
4k views

Equation of a line passing through a given point, perpendicular with a vector

Find the line that goes through A(1,0,2) and is perpendicular to r = (-2,3,4) + s (1,1,2) I did a bunch of work, but I don't know if any of it is right. I erased most of it, but this is what I came ...
2
votes
1answer
2k views

Convert two points to line eq (Ax + By +C = 0)

Say one has two points in the x,y plane. How would one convert those two points to a line? Of course I know you could use the slope-point formula & derive the line as following: $$y - y_0 = ...
0
votes
2answers
111 views

Coordinate Geometry Problem

I have a question that I've started at school but had couldn't figure out what to do or where to start. Sorry, I don't have the question written down, just the image. ...
0
votes
1answer
71 views

Generating Vectors under Constraints on 1 and 2 norm

Update: I left out some important information in my previous description... I am actually dealing with a special problem, which is better described as follows: Given user-specified parameters ...
1
vote
1answer
151 views

A line and a plane in 3 dimensions

Line L is defined as point P(3, 2, -2) with a directional vector v <1, -1, 2>. Plane S: A (1, 2, 1), B (2, -1, 2), C (0, -2, 1) a) When t = 2, how far away is the particle's position located from ...
9
votes
5answers
232 views

How to think of $\vec{u}-\vec{v}$

Assume I have two vectors, $\vec{u}$ and $\vec{v}$. I know that I can think of their sum via Triangle or Parallelogram Law, but I'm having trouble knowing which way the vector would point depending on ...
1
vote
1answer
580 views

finding the coordinates of a point of intersection: 3d sphere and plane

How to find the coordinates of one point on the interaction of the sphere $$(x-1)^2+(y-2)^2+(z-4)^2= 25$$ and the plane $z=4$. I was trying to solve this I got it down to $x+y=8$ but then when I ...
1
vote
3answers
360 views

Arc Length Formulas

Use the arc length formula to find the arc length of the upper half of the circle with center at $(0,0)$ and radius $3$. Also, find the arc length of the curve in the first question by using ...
2
votes
4answers
97 views

describe the domain of a function $f(x, y)$

Describe the domain of the function $f(x,y) = \ln(7 - x - y)$. I have the answer narrowed down but I am not sure if it would be $\{(x,y) \mid y ≤ -x + 7\}$ or $\{(x,y) \mid y < -x + 7\}$ please ...
1
vote
0answers
187 views

Canonical form of a curve (geometry)

I am bothering with this geometric problem more than half a day and couldn't understand it yet. Here it is: In orthonormal coordinate system K=Oxy we have a curve C: $9x^2 - 4xy + 6y^2 + 6x - 8y + ...
0
votes
3answers
8k views

Find the equation of a circle given two points and a line that passes through its center

Find the equation of the circle that passes through the points $(0,2)$ and $(6,6)$. Its center is on the line $x-y =1$.
-1
votes
3answers
56 views

given point (2,6) and a line passes through point (3,0)

The question is: does the distance between the point $(2,6)$ to the that line could be $5$? is there a solution to the problem without computing? i would glad to know. thanks.
2
votes
2answers
550 views

How to find the shortest path between two points under the restriction?

Let two different points $M_1(x_1,y_1,z_1)$, $M_2(x_2,y_2,z_2)$ and two nonintersecting lines $l_1$, $l_2$ be given. How to find the shortest path between $M_1$ and $M_2$ which intersects both $l_1$ ...
2
votes
1answer
95 views

3D Geometry Question

In $3$-dimensional Geometry, if angle made of line segment $OP$ with $X,Y,Z$-axis are in $1:2:3$, then what is the angle made by line segment with $Y$-axis? My Solution: Let $\alpha,\beta$ and ...
0
votes
3answers
327 views

Quadratic equation with parameter

Stuck solving this equation. Full text: For what real values of the parameter do the common solutions of the equation became identical? ...
0
votes
2answers
143 views

How many times can quadric kiss cosine at given point?

Let a quadric $ax^2+2bxy+cy^2+dx+ey+f=0$ touches the plot of $y=\cos(x)$ at the point $(0,1)$ with multiplicity $n$. What is the maximum possible value of $n$? Recall that a joint point $P$ of ...
0
votes
1answer
63 views

Necessary equation for envelopes

Consider a family of curves parameterised by $t$, where each member of the family is described by $$F(t,x,y) = 0.$$ Define an envelope of the family to be a curve $E$ such that every point on $E$ is ...
1
vote
2answers
77 views

Being inside or outside of an ellipse

Let $A$ be a point $A$ not belonging to an ellipse $E$. We say that $A$ lies inside $E$ if every line passing trough $A$ intersects $E$. We say that $A$ lies otside $E$ if some line passing trough $A$ ...
0
votes
1answer
82 views

Measuring distances on any coordinate system

I was reading the book The ABC of Relativity from Betrand Russell, and at some point, the author mentions a method for measuring the distance between 2 points on any coordinate system. He says that ...
0
votes
1answer
58 views

analytical geometry

We have an affine coordinate system and $3$ points given: $A=(1,0,0)$, $B=(0,1,0)$, $C=(0,0,1)$, $D=(1,1,1)$. I have to find a linear transformation, which depicts the points $A$, $B$, $C$ and $D$ ...
2
votes
1answer
164 views

how to dot product two vectors with different planes?

how to dot product two vectors with different planes? I have vectors $A$,$B$ and $C$, vectors $A$ and $B$ is on $xy$ plane while vector $C$ is on $xz$ plane. I need to find the dot product of $A.C$ ...
0
votes
1answer
127 views

Geometric question?

First of all, is it Geometric? Image of the drafted: I need help solving this question, and I am completely lost on how can I solve this. Could anyone explain the way of solving this geometric ...
1
vote
0answers
71 views

How to introduce perpendicular or congruence of angles in affine space

$n$-dimensional affine point-vector space is a pair $\mathbb A^n = \langle \mathbb A, V^n \rangle$, where $\mathbb A$ is an arbitrary set, which elements are called points of affine space, $V^n$ is an ...
1
vote
1answer
68 views

Find the transform

I have the paper with 3 points on it. I have also a photo of this paper. How can I determine where is the paper on the photo, if I know just the positions of these points? And are 3 points enough? It ...
0
votes
1answer
102 views

triangle, vectors, proving an identity.

I'm trying to prove something but unfortunately I can't. Let $ABC$ be a triangle and $M$ a point in $[AB]$ where $d(A,M)=d(B,M)$.Let also be $N$ be a point in $[AC]$ where $d(A,N)=d(B,N)$. Prove ...
0
votes
1answer
50 views

Expressing a point in two coordinate systems

Let $(O,e_1,e_2,e_3)$ and $(O',e_1',e_2',e_3')$ be two coordinate systems. Let $\overline{OO'}=2e_1-e_2+3e_3$, $e'_1=e_1-e_2+3e_3$, $e'_2=e_1+e_2+e_3$ and $e'_3=e_1-e_2-e_3$. a) Find the coordinates ...
0
votes
2answers
157 views

The closes point to a curve in space.

I am working on the following problem. Find the point closest to the origin, of the curve of intersection of the plane $2y+4z =5$ and the cone $z^2 = 4(x^2+y^2)$ I was able to see that the ...
1
vote
0answers
125 views

Describing domain of integration of triple integral

I'm struggling to visualize the following problem: This question concerns the integral $\int_{0}^{2}\int_0^{\sqrt{4-y^2}}\int_{\sqrt{x^2+y^2}}^{\sqrt{8-x^2-y^2}}\!z\ \mathrm{d}z\ \mathrm{d}x\ ...
1
vote
1answer
1k views

Find equations of the ellipses given conditions on the directrices, foci, and vertices

The ellipses have their centers at the origin and their major axes on the $x$-axis. Find the equation: with distance between directrices $27$, and between foci $3$; with a focus at $(-\sqrt{13},0)$ ...
0
votes
2answers
88 views

Finding Vectors in cartesian form

I am stuck on this question could you please help me. Find,in Cartesian form, the equations of the straight line through the point with position vector (-1,2,-3) parallel to the direction given by ...
1
vote
1answer
526 views

Line equation - parametric and canonical

Let's say I have a line in R3: $$ l:\begin{cases} x-3y+3z=0\\ x+2y-2z=2 \end{cases} $$ How to change it to canonical and parametric equation?
2
votes
2answers
491 views

To use the two-point formula to find the linear equation relating $C$ and $F$:

I've tried to solve a problem which I'm going to give below. What I don't understand is that which variable is dependent and which is independent among $C$ and $F$. I think we can relate $C$ and $F$ ...
2
votes
3answers
2k views

To find the x and y-intercepts of the line $ax+by+c=0$

Please check if I've solved the problem in the correct way: The problem is as follows: Find the points at which the line $ax+by+c=0$ crosses the x and y-axes. (Assume that $a \neq 0$ and $b \neq ...
3
votes
4answers
133 views

Equation of the line that has $x$ and $y$ intercepts at $a$ and $b$.

Please can anyone help me with proving the following problem: Show that the line that crosses the $X$-axis at $a \neq 0$ and the $Y$-axis at $b \neq 0$ has the equation $$\dfrac{x}a + \dfrac{y}b ...
2
votes
1answer
200 views

Proof that differential of differential form $=0$ i.e $d(df) = 0$.

Let $f$ be a differentiable function on an open space $U \subset \mathbb{R}^n$. Proove that $d(df) = 0$. So my proof is: Let $$f = \sum c_{i_1, \cdots i_k}(x_*)dx_{i_1} \wedge \cdots ...
0
votes
1answer
52 views

Prove that the $2$ form defines a symplectic structure

Prove that the $2$ form $$\omega = -2[(1+x_2^2)dx_1 \wedge dx_2 + dx_1 \wedge dx_3 + dx_3 \wedge dx_4]$$ defines a symplectic structure on $\mathbb{R}_x^4$. My definition of as ...
5
votes
0answers
206 views

Are there eigenvectors, eigenvalues, and characteristic functions for non-linear vector fields?

An eigenvector is a vector in the preimage of the transformation whose direction is not changed when the transformation is applied. It seems like the concept of eigenvectors and eigenvalues would ...
1
vote
2answers
248 views

How can we derive the standard form of the linear equation: $Ax+By+C$?

How can we derive the standard form of the linear equation: $Ax+By+C=0$? What do "$A$", "$B$" and "$C$" in the standard form of the linear equation mean? As in the point-slope form of the linear ...
0
votes
2answers
6k views

Find the equation of a locus…(Read More)

Find the equation of the locus of a point which moves so that it's distance from (4,-3) is always one-half its distance from (-1,-1).
0
votes
3answers
56 views

Identify the curve with the following equation.

To "identify" means not only to name but to give pertinent data, such as center, foci and axes, if they exist. $$4x^2=4x-y^2$$
1
vote
2answers
215 views

Local Diffeomorphism Theoerm

Is this correct for the local diffeomorphism theorem: A multivariable function $F(x_1, \cdots x_n)$ has a local diffeomorphism at a point $a = (a_1, \cdots a_n)$ if the determinant of the Jacobian ...