Questions on the use of algebraic techniques to prove geometric theorems.
1
vote
2answers
44 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 $A$ ...
15
votes
1answer
312 views
Intuition why the volume and surface area of the unit sphere eventually decrease
The volume formula for a unit sphere, $$\frac{\pi^{n/2}}{\Gamma{(1 + n/2)}},$$
and the surface area formula, $$\frac{2\pi^{n/2}}{\Gamma{(n/2)}},$$
both attain maximum values for finite $n$. We can ...
0
votes
1answer
33 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
32 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 $D$ ...
4
votes
0answers
105 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 ...
0
votes
1answer
92 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 ...
0
votes
0answers
25 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 ...
0
votes
1answer
32 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
28 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
165 views
Can you suggest me a good book for self-study of analytic geometry 1?
I'm stuyding mathematics alone, but I plan to enter in the university in the near future, I went to the university website and it suggests Analytic Geometry 1 as part of their curricula, the topics ...
2
votes
1answer
1k views
Getting the third point from two points on one line
My question is the following
how can i get the x3,y3 point from x1,y1 and x2,y2 points?
the x3,y3 point distance from x1,y1 is 300.
0
votes
1answer
28 views
Get the equation for a plane when we know a point and an intersection between two planes
The point is $P:(1,4,-2)$ and the two planes that the equation intersect is $$\pi_1:2x+2y-z+4=0$$$$\pi_2:3x-y+3z+1=0$$ what is the equation?
1
vote
1answer
29 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
32 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 ...
1
vote
1answer
266 views
Analytic proof for Circles of Apollonius
I'm looking for an analytic proof the statement for a Circle of Apollonius (I found a geometrical one already): If $\overline{AC}:\overline{BC}=s$, then $P \in k_s$. $s \in (0,1)$.
$k_s$ is the ...
-1
votes
0answers
25 views
A problem of Analytical Geometry.
Lines are drawn through the origin to meet the circle in which plane $x+y+z=1$ cuts the sphere $x^2+y^2+z^2-4x-6y-8z+4=0$. How to show that they meet the sphere again at points on the plane $y+2z=2$?
0
votes
0answers
40 views
How to prove the property of scalar distribution over vector addition when the vectors are collinear?
$\overrightarrow{a},\overrightarrow{b} \in V^3 , \alpha \in \mathbb{R} $
Prove: $\alpha(\overrightarrow{a} +\overrightarrow{b}) = \alpha\overrightarrow{a} + \alpha\overrightarrow{b}$
When $\alpha = ...
1
vote
1answer
19 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?
1
vote
2answers
47 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 ...
2
votes
3answers
51 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 ...
2
votes
2answers
37 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 $F$ ...
3
votes
4answers
54 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 $$\dfrac{x}a + \dfrac{y}b ...
2
votes
1answer
46 views
Proof that differential of differential form $=0$ i.e $d(df) = 0$.
Let $f$ be a differentiable function on an open space $U \subset \mathbb{R}^n$. Proove that $d(df) = 0$.
So my proof is:
Let
$$f = \sum c_{i_1, \cdots i_k}(x_*)dx_{i_1} \wedge \cdots ...
0
votes
1answer
23 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 ...
0
votes
2answers
35 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).
5
votes
5answers
514 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 ...
0
votes
3answers
30 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
131 views
Any video lectures on conventional analytical geometry?
Hi this question is kind of a natural offshoot to this question My topic covers very conventional topics like :
Cartesian and Polar Coordinates in 3 Dim, second Degree eqns in 3 vars, reduction ...
5
votes
1answer
370 views
Geometry IMO 1988
(IMO 1988/1) Consider two circles of radii $R$ and $r$ $(R > r)$ with the same center. Let $P$ be a fixed point on the smaller circle and $B$ a variable point on the larger circle. The line $BP$ ...
0
votes
0answers
51 views
Derivation of $Ax+By+C=0$
Below I've tried to derive the standard form of the linear equation. Please check my derivation.
Object: Derive $Ax+By+C=0$.
The slope-intercept form of the linear equation can be given as:
...
-1
votes
1answer
30 views
Solve a parabola problem?
So, two parabolas are given: $y^2=24x$ and $x^2=3y$ and a point $A(24,3)$. If B and O are intersect points of these two parabolas, prove that the angle ABO is right.
1
vote
1answer
36 views
Coordinates and locus of centroid
A triangle has two of its sides along the co-ordinate axes and its third side is a tangent to the circle $x^2+y^2=a^2$. If the coordinates of the point of contact of the tangent are $(a \cosØ,a ...
0
votes
1answer
42 views
How can you use the parametric form of a plane in the following application?
With regards to computer graphics, how can you make use of the parametric representation of a plane and how of its normal form?
1
vote
1answer
43 views
Willmore energy of an ellipsoid
Given an ellipsoid of equation
$$\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$$
How can I calculate the Willmore energy of this surface knowing that its definition is:
...
1
vote
1answer
59 views
Triangle optimization problem
Let $a,b,c$ be the sides of a triangle , then what is the maximum and minimum values (if exist) of the following quantities
(i) $\dfrac {a^2b^2c^2}{(a+2b)(a+2c)(b+2c)(b+2a)(c+2a)(c+2b)}$
(ii) ...
11
votes
2answers
294 views
When do equations represent the same curve?
Suppose we have two sets of parametric equations $\mathbf c_1(u) = (x_1(u), y_1(u))$ and $\mathbf c_2(v) = (x_2(v), y_2(v))$ representing two 2D planar curves. When I say "2D planar curves" I mean ...
0
votes
0answers
71 views
Overlapping Areas
Knowing the areas of A, B and C, is there an analytical way to find out if two arbitrary shapes overlap each other in the plane? (See image here: http://i.imgur.com/RWsqysT.jpg)
More formally, I'm ...
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 ...
1
vote
4answers
42 views
find vector reflected across another vector
I apologize that this may sound like a very basic question, but I can't find any clear answers in my search. I have a vector $\vec{v}$ that I want to reflect across a vector $\vec{n}$. The dot product ...
1
vote
1answer
39 views
Understanding the graph of the displacement of a particle wirh respect to time
At time $t=0$ the position of the particle is $3 ft$, and at time $t=2$ the position of the particle is $11ft$. At time $t=0$ the velocity of the particle must have been zero. So if its the motion ...
3
votes
2answers
53 views
Slope; A measure of Direction
In my book, the definition of the slope of the straight line is:
The slope is a measure of the direction of the line.
1) When the line has no slope, it tells that it is vertical or moving ...
2
votes
3answers
65 views
Analytic geometry straight line problem
Prove that two straight lines represented by the equation $x^3+y^3+bx^2y+cxy^2=0$ will be at right angles if $b+c=-2$
I didn't know that even straight lines like planes can be represented by a ...
0
votes
2answers
28 views
Analytic geometry section of cone and sphere
How to show that the cone $yz+zx+xy=0$ cuts the sphere $x^2+y^2+z^2=a^2$ in two equal circle ?
I understand that the two equations taken together represent the circle. but how to go about finding the ...
1
vote
3answers
76 views
surface area of a sphere above a cylinder
I need to find the surface area of the sphere $x^2+y^2+z^2=4$ above the cone $z = \sqrt{x^2+y^2}$, but I'm not sure how. I know that the surface area of a surface can be calculated with the equation ...
3
votes
1answer
153 views
Algebra question about Triangle Interiors
I was reading about Triangle Interiors on Wolfram Alpha:
http://mathworld.wolfram.com/TriangleInterior.html
and they have a simple equation:
$$\mathbf{v} = \mathbf{v}_0 + a\mathbf{v}_1 + ...
0
votes
2answers
61 views
Straight Line Definition
Definition: Let $P(x_0, y_0)$ be a point and let $m$ be a real number. The line through $P$ with slope $m$ is the set of all points $Q(x, y)$ with,
$y -y_0 = m(x - x_0)$
Does the set of all ...
2
votes
4answers
1k views
How to know if a point is inside a circle?
Having a circle with the centre $(x_c, y_c)$ with the radius $r$ how to know whether a point $(x_p, y_p)$ is inside the circle?
3
votes
1answer
25 views
Point-slope Equation
Let suppose that $R(x_1, y_1)$ is a point on the (x, y)plane and a line $L$ with slope $m$ passes through this point. There is a point $S(x_1, y_1)$ on $L$ such that $R$ and $S$ are coincident points. ...
0
votes
2answers
32 views
Slope of a straight line
Why is this so that a higher value of slope indicates a steeper incline? I can't take it into my head. What could be the reason behind that? I know that it is a fact because I've also noticed it but ...
2
votes
3answers
57 views
Distance between point and line in the complex plane
Let $a,b$ be fixed complex numbers and let $L$ be the line
$$L=\{a+bt:t\in\Bbb R\}.$$
Let $w\in\Bbb C\setminus L$. Let's calculate $$d(w,L)=\inf\{|w-z|:z\in L\}=\inf_{t\in\Bbb R}|w-(a+bt)|.$$
The ...
